We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equ... more We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation ∂tu − ∆u = |u| α u, α > 0 with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular, sign-changing, as well as non polynomial decaying initial data are considered. The proofs of the lower-bound estimates of lifespan are based on the local construction of solutions. The proofs of the upper-bounds exploit a well-known necessary condition for the existence of nonnegative solutions. In addition, we establish new results for lifespan using dilation methods and we give new lifespan estimates for Hardy-Hénon parabolic equations.
In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation ∂tu... more In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation ∂tu+∆ 2 u = f (u), t > 0, x ∈ R N , with f (u) ∼ e u 2 for large u. Under smallness condition on the initial data and for exponential nonlinearity f such that |f (u)| ∼ |u| m as u → 0, m ≥ 2, N (m − 1)/4 ≥ 2, we show that the solution is global. Moreover, we obtain decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.
In this paper we consider the inhomogeneous nonlinear Schrödinger equation i∂tu + ∆u = K(x)|u| α ... more In this paper we consider the inhomogeneous nonlinear Schrödinger equation i∂tu + ∆u = K(x)|u| α u, u(0) = u0 ∈ H s (R N), s = 0, 1, N ≥ 1, |K(x)| + |x| s |∇ s K(x)| |x| −b , 0 < b < min(2, N − 2s), 0 < α < (4 − 2b)/(N − 2s). We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted L 2-space for a new range α0(b) < α < (4 − 2b)/N. The value α0(b) is the positive root of N α 2 + (N − 2 + 2b)α − 4 + 2b = 0, which extends the Strauss exponent known for b = 0. Our results improve the known ones for K(x) = µ|x| −b , µ ∈ C and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of α. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential K. Here u = u(t, x) ∈ C, t ∈ R, x ∈ R N , N ≥ 1, s = 0 or s = 1 and α > 0. The potential K is a complex valued function satisfying some hypothesis. In particular, K(x) = µ|x| −b and K(x) = µ(1 + |x| 2) − b 2 , b > 0, µ ∈ C, will be considered. Equation (1.1) with a constant function K, corresponds to the standard nonlinear Schrödinger equation. The case where K is non constant and bounded is considered in [33, 35]. The unbounded potential case is also treated in [11, 12, 41], where K(x) = |x| b. Here we consider (1.1) with potential having a decay like |x| −b at infinity and may be singular at the origin. This kind
We investigate the existence and the boundary regularity of source-type self-similar solutions to... more We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation ht = −(h n hzzz)z + (h n+3)zz, t > 0, z ∈ R; h(0, z) = ωδ where n ∈ (3/2, 3), ω > 0 and δ is the Dirac mass at the origin. It is known that the leading order expansion near the edge of the support coincides with that of a travelling-wave solution for the standard thin-film equation: ht = −(h n hzzz)z. In this paper we sharpen this result, proving that the higher order corrections are analytic with respect to three variables: the first one is just the spacial variable, whereas the second and third (except for n = 2) are irrational powers of it. It is known that this third order term does not appear for the thin-film equation without gravity.
Proceedings of the International Congress of Mathematicians (ICM 2018), 2019
In this paper we consider the problem: ∂tu − ∆u = f (u), u(0) = u0 ∈ exp L p (R N), where p > 1 a... more In this paper we consider the problem: ∂tu − ∆u = f (u), u(0) = u0 ∈ exp L p (R N), where p > 1 and f : R → R having an exponential growth at infinity with f (0) = 0. We prove local well-posedness in exp L p 0 (R N) for f (u) ∼ e |u| q , 0 < q ≤ p, |u| → ∞. However, if for some λ > 0, lim inf s→∞ f (s) e −λs p > 0, then non-existence occurs in exp L p (R N). Under smallness condition on the initial data and for exponential nonlinearity f such that |f (u)| ∼ |u| m as u → 0, N(m−1) 2 ≥ p, we show that the solution is global. In particular, p − 1 > 0 sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on m.
In this paper we consider the initial value problem ∂tu − ∆u = f (u), u(0) = u0 ∈ exp L p (R N), ... more In this paper we consider the initial value problem ∂tu − ∆u = f (u), u(0) = u0 ∈ exp L p (R N), where p > 1 and f : R → R having an exponential growth at infinity with f (0) = 0. Under smallness condition on the initial data and for nonlinearity f such that |f (u)| ∼ e |u| q as |u| → ∞, |f (u)| ∼ |u| m as u → 0, 0 < q ≤ p ≤ m, N(m−1) 2 ≥ p > 1, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on m.
We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation \begin{docume... more We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation \begin{document}$ i\partial_t u +\Delta u = \mu |x|^{-b}|u|^\alpha u,\; u(0)\in H^s({\mathbb R}^N),\; N\geq 1,\; \mu\in {\mathbb C},\; \; b>0 $\end{document} and \begin{document}$ \alpha>0. $\end{document} Only partial results are known for the local existence in the subcritical case \begin{document}$ \alpha<(4-2b)/(N-2s) $\end{document} and much more less in the critical case \begin{document}$ \alpha = (4-2b)/(N-2s). $\end{document} In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for ...
Journal of Mathematical Analysis and Applications, 2020
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
In this paper, we study global well-posedness and long-time asymptotic behavior of solutions to t... more In this paper, we study global well-posedness and long-time asymptotic behavior of solutions to the nonlinear heat equation with absorption, u t - Δ u + | u | α u = 0 {u_{t}-\Delta u+\lvert u\rvert^{\alpha}u=0} , where u = u ( t , x ) ∈ ℝ {u=u(t,x)\in\mathbb{R}} , ( t , x ) ∈ ( 0 , ∞ ) × ℝ N {(t,x)\in(0,\infty)\times\mathbb{R}^{N}} and α > 0 {\alpha>0} . We focus particularly on highly singular initial values which are antisymmetric with respect to the variables x 1 , x 2 , … , x m {x_{1},x_{2},\ldots,x_{m}} for some m ∈ { 1 , 2 , … , N } {m\in\{1,2,\ldots,N\}} , such as u 0 = ( - 1 ) m ∂ 1 ∂ 2 ⋯ ∂ m | ⋅ | - γ ∈ 𝒮 ′ ( ℝ N ) {u_{0}=(-1)^{m}\partial_{1}\partial_{2}\cdots\partial_{m}\lvert\,{\cdot}\,% \rvert^{-\gamma}\in\mathcal{S}^{\prime}(\mathbb{R}^{N})} , 0 < γ < N {0<\gamma 0 {\alpha>0} . Our approach is to study well-posedness and large time behavior on sectorial domains of the form Ω m = { x ∈ ℝ N : x 1 , … , x m > 0 } {\Omega_{m}=\{x\in\mathbb{R}^{N}:...
Transactions of the American Mathematical Society, 2019
We consider a nonlinear heat equation with a double source: | u | p − 1 u |u|^{p-1}u and | ∇ u | ... more We consider a nonlinear heat equation with a double source: | u | p − 1 u |u|^{p-1}u and | ∇ u | q |\nabla u|^q . This equation has a double interest: in ecology, it was used by Souplet (1996) as a population dynamics model; in mathematics, it was introduced by Chipot and Weissler (1989) as an intermediate equation between the semilinear heat equation and the Hamilton-Jacobi equation. Further interest in this equation comes from its lack of variational structure. In this paper, we intend to see whether the standard blow-up dynamics known for the standard semilinear heat equation (with | u | p − 1 u |u|^{p-1}u as the only source) can be modified by the addition of the second source ( | ∇ u | q |\nabla u|^q ). Here arises a nice critical phenomenon at blow-up: - when q > 2 p / ( p + 1 ) q>2p/(p+1) , the second source is subcritical in size with respect to the first, and we recover the classicial blow-up profile known for the standard semilinear case; - when q = 2 p / ( p + 1 ) q...
In this paper we prove local existence of solutions to the nonlinear heat equation ut = ∆u + a|u|... more In this paper we prove local existence of solutions to the nonlinear heat equation ut = ∆u + a|u| α u, t ∈ (0, T), x = (x1, • • • , xN) ∈ R N , a = ±1, α > 0; with initial value u(0) ∈ L 1 loc R N \ {0} , anti-symmetric with respect to x1, x2, • • • , xm and |u(0)| ≤ C(−1) m ∂1∂2 • • • ∂m(|x| −γ) for x1 > 0, • • • , xm > 0, where C > 0 is a constant, m ∈ {1, 2, • • • , N }, 0 < γ < N and 0 < α < 2/(γ + m). This gives a local existence result with highly singular initial values. As an application, for a = 1, we establish new blowup criteria for 0 < α ≤ 2/(γ + m), including the case m = 0. Moreover, if (N − 4)α < 2, we prove the existence of initial values u0 = λf, for which the resulting solution blows up in finite time Tmax(λf), if λ > 0 is sufficiently small. We also construct blowing up solutions with initial data λnf such that λ [(1 α − γ+m 2) −1 ] n Tmax(λnf) has different finite limits along different sequences λn → 0. Our result extends the known "small lambda" blow up results for new values of α and a new class of initial data.
Proceedings of the American Mathematical Society, 2017
We show the uniqueness of strong solutions for the thin-film equation ut + (uuxxx)x = 0 with init... more We show the uniqueness of strong solutions for the thin-film equation ut + (uuxxx)x = 0 with initial data u(0) = mδ, m > 0 where δ is the Dirac mass at the origin. In particular, the solution is the source type one obtained by Smyth and Hill. The argument is based on an entropy estimate for the equation in self-similar variables.
Nonlinear Analysis: Theory, Methods & Applications, 2017
In this paper we study the nonlinear parabolic equation ∂tu = ∆u + a|x| −γ |u| α u, t > 0, x ∈ R ... more In this paper we study the nonlinear parabolic equation ∂tu = ∆u + a|x| −γ |u| α u, t > 0, x ∈ R N \{0}, N ≥ 1, a ∈ R, α > 0, 0 < γ < min(2, N) and with initial value u(0) = ϕ. We establish local well-posedness in L q (R N) and in C0(R N). In particular, the value q = N α/(2 − γ) plays a critical role. For α > (2 − γ)/N, we show the existence of global self-similar solutions with initial values ϕ(x) = ω(x)|x| −(2−γ)/α , where ω ∈ L ∞ (R N) is homogeneous of degree 0 and ω ∞ is sufficiently small. We then prove that if ϕ(x) ∼ ω(x)|x| −(2−γ)/α for |x| large, then the solution is global and is asymptotic in the L ∞-norm to a self-similar solution of the nonlinear equation. While if ϕ(x) ∼ ω(x)|x| −σ for |x| large with (2 − γ)/α < σ < N, then the solution is global but is asymptotic in the L ∞-norm to e t∆ (ω(x)|x| −σ). The equation with more general potential, ∂tu = ∆u + V (x)|u| α u, V (x)|x| γ ∈ L ∞ (R N), is also studied. In particular, for initial data ϕ(x) ∼ ω(x)|x| −(2−γ)/α , |x| large , we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity.
We study positive blowing-up solutions of systems of the form: ut = δ1∆u + e pv , vt = δ2∆v + e q... more We study positive blowing-up solutions of systems of the form: ut = δ1∆u + e pv , vt = δ2∆v + e qu , with δ1, δ2 > 0 and p, q > 0. We prove single-point blow-up for large classes of radially decreasing solutions. This answers a question left open in a paper of Friedman and Giga (1987), where the result was obtained only for the equidiffusive case δ1 = δ2 and the proof depended crucially on this assumption.
Nous etudions les solutions autosimilaires a symetrie radiale de certaines equations semi-lineair... more Nous etudions les solutions autosimilaires a symetrie radiale de certaines equations semi-lineaires paraboliques avec terme de gradient non lineaire. La these se divise en deux parties independantes. Dans la premiere partie nous demontrons l'existence de solutions autosimilaires definies pour le temps positif, qui sont globales, strictement positives et avec donnees initiales singulieres a l'origine. En particulier, ceci conduit a un resultat de non unicite pour le probleme de Cauchy associe a ces equations dans certains espaces de Lebesgue et de Sobolev. Dans la deuxieme partie, nous considerons le cas unidimensionnel et de plus le gradient est affecte d'une constante negative. Nous demontrons l'existence d'une solution autosimilaire non triviale definie pour le temps negatif. En particulier, ceci implique l'existence d'une solution explosive dont l'ensemble des points d'explosion est reduit a un seul point. Le profil final de cette solution au moment de l'explosion et au voisinage de ce point est different de tous les profils possibles, dans le cas d'un seul point d'explosion, pour cette equation sans terme de gradient, a savoir l'equation de la chaleur avec terme de source
We show that the heat semigroup is well defined on the Banach space $${\mathcal{X}_{m,\gamma} = \... more We show that the heat semigroup is well defined on the Banach space $${\mathcal{X}_{m,\gamma} = \{ \psi:\Omega_m \to \mathbb{R} ;\; |x|^{\gamma +2m}(\prod_{i=1}^m x_i)^{-1}\psi(x) \in L^\infty(\Omega_m)\},}$$Xm,γ={ψ:Ωm→R;|x|γ+2m(∏i=1mxi)-1ψ(x)∈L∞(Ωm)},$${0 < \gamma < N}$$0 0,\, 1\leq i\leq m\},}$$Ωm={(x1,x2,…,xN)∈RN;xi>0,1≤i≤m},$${1\leq m\leq N}$$1≤m≤N. We then investigate the large time behavior of solutions of the heat equation $${u_{t}-\Delta u=0}$$ut-Δu=0 for t > 0 and $${x \in \Omega_m.}$$x∈Ωm. Using certain notions from dynamical systems, we show that the large time behavior is related to the spatial asymptotic behavior of its initial value. Since the space $${\mathcal{X}_{m, \gamma}}$$Xm,γ contains highly singular initial data, which can be extended to all of $${\mathbb{R}^{N}}$$RN by antisymmetry, we also obtain new results on the complexity in the asymptotic behavior of solutions for the heat equation on the whole space.
We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat ... more We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation: ut = D 2α x u ∓ u 2 , t ∈ (0, T), x ∈ R or T, with 0 < α ≤ 1 is well-posed in H s for s ≥ max(−α, 1/2 − 2α) except in the case α = 1/2 where it is shown to be well-posed for s > −1/2 and ill-posed for s = −1/2. As a by-product we improve the known well-posedness results for the heat equation (α = 1) by reaching the end-point Sobolev index s = −1. Finally, in the case 1/2 < α ≤ 1, we also prove optimal results in the Besov spaces B s,q 2 .
We study positive blowing-up solutions of the system: ut − δ∆u = v p , vt − ∆v = u q , as well as... more We study positive blowing-up solutions of the system: ut − δ∆u = v p , vt − ∆v = u q , as well as of some more general systems. For any p, q > 1, we prove single-point blow-up for any radially decreasing, positive and classical solution in a ball. This improves on previously known results in 3 directions: (i) no type I blow-up assumption is made (and it is known that this property may fail); (ii) no equidiffusivity is assumed, i.e. any δ > 0 is allowed; (iii) a large class of nonlinearities F (u, v), G(u, v) can be handled, which need not follow a precise power behavior. As side result, we also obtain lower pointwise estimates for the final blow-up profiles.
Nonlinear Differential Equations and Applications NoDEA, 2015
In this paper we study the existence and the asymptotic behavior of global solutions for a parabo... more In this paper we study the existence and the asymptotic behavior of global solutions for a parabolic system related to the complexvalued heat equation with quadratic nonlinearity: ∂tz = Δz + z 2 , t > 0, x ∈ R N , with initial data z0 = u0 + iv0. We show that if u0(x) ∼ c|x| −2α 1 and v0(x) ∼ c|x| −2α 1 , as |x| → ∞ with α1 ≥ 1, 2α 1 − α1 ≥ 1, N 2α 1 > 1, N 2α 1 > 1 (|c| is sufficiently small), then the solution is global and converges to a self-similar solution. We also establish the existence of four different self-similar behaviors. These behaviors depend on the values of α1 and α 1. In particular, the real and the imaginary parts of the constructed solutions may have different behaviors in the L ∞-norm for large time. Also, the real part may have different behaviors from those known for the real-valued quadratic heat equation.
Transactions of the American Mathematical Society, 2013
In this paper we prove local existence of solutions of the nonlinear heat equation u t = Δ u + | ... more In this paper we prove local existence of solutions of the nonlinear heat equation u t = Δ u + | u | α u , t ∈ ( 0 , T ) , x ∈ R N , u_t = \Delta u + |u|^\alpha u, \; t\in (0,T),\; x\in \mathbb {R}^N, with initial value u ( 0 ) = K ∂ 1 ∂ 2 ⋅ ⋅ ⋅ ∂ m δ , K ≠ 0 , m ∈ { 1 , 2 , ⋯ , N } , 0 > α > 2 / ( N + m ) u(0)=K\partial _{1}\partial _{2}\cdot \cdot \cdot \partial _{m}\delta ,\; K\not =0,\; m\in \{1,\; 2,\; \cdots ,\; N\},\; 0>\alpha >2/(N+m) and δ \delta is the Dirac distribution. In particular, this gives a local existence result with an initial value in a high order negative Sobolev space H s , q ( R N ) H^{s,q}(\mathbb {R}^N) with s ≤ − 2. s\leq -2. As an application, we prove the existence of initial values u 0 = λ f u_0 = \lambda f for which the resulting solution blows up in finite time if λ > 0 \lambda >0 is sufficiently small. Here, f f satisfies in particular f ∈ C 0 ( R N ) ∩ L 1 ( R N ) f\in C_0(\mathbb {R}^N)\cap L^1(\mathbb {R}^N) and is anti-symmetri...
We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equ... more We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation ∂tu − ∆u = |u| α u, α > 0 with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular, sign-changing, as well as non polynomial decaying initial data are considered. The proofs of the lower-bound estimates of lifespan are based on the local construction of solutions. The proofs of the upper-bounds exploit a well-known necessary condition for the existence of nonnegative solutions. In addition, we establish new results for lifespan using dilation methods and we give new lifespan estimates for Hardy-Hénon parabolic equations.
In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation ∂tu... more In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation ∂tu+∆ 2 u = f (u), t > 0, x ∈ R N , with f (u) ∼ e u 2 for large u. Under smallness condition on the initial data and for exponential nonlinearity f such that |f (u)| ∼ |u| m as u → 0, m ≥ 2, N (m − 1)/4 ≥ 2, we show that the solution is global. Moreover, we obtain decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.
In this paper we consider the inhomogeneous nonlinear Schrödinger equation i∂tu + ∆u = K(x)|u| α ... more In this paper we consider the inhomogeneous nonlinear Schrödinger equation i∂tu + ∆u = K(x)|u| α u, u(0) = u0 ∈ H s (R N), s = 0, 1, N ≥ 1, |K(x)| + |x| s |∇ s K(x)| |x| −b , 0 < b < min(2, N − 2s), 0 < α < (4 − 2b)/(N − 2s). We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted L 2-space for a new range α0(b) < α < (4 − 2b)/N. The value α0(b) is the positive root of N α 2 + (N − 2 + 2b)α − 4 + 2b = 0, which extends the Strauss exponent known for b = 0. Our results improve the known ones for K(x) = µ|x| −b , µ ∈ C and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of α. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential K. Here u = u(t, x) ∈ C, t ∈ R, x ∈ R N , N ≥ 1, s = 0 or s = 1 and α > 0. The potential K is a complex valued function satisfying some hypothesis. In particular, K(x) = µ|x| −b and K(x) = µ(1 + |x| 2) − b 2 , b > 0, µ ∈ C, will be considered. Equation (1.1) with a constant function K, corresponds to the standard nonlinear Schrödinger equation. The case where K is non constant and bounded is considered in [33, 35]. The unbounded potential case is also treated in [11, 12, 41], where K(x) = |x| b. Here we consider (1.1) with potential having a decay like |x| −b at infinity and may be singular at the origin. This kind
We investigate the existence and the boundary regularity of source-type self-similar solutions to... more We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation ht = −(h n hzzz)z + (h n+3)zz, t > 0, z ∈ R; h(0, z) = ωδ where n ∈ (3/2, 3), ω > 0 and δ is the Dirac mass at the origin. It is known that the leading order expansion near the edge of the support coincides with that of a travelling-wave solution for the standard thin-film equation: ht = −(h n hzzz)z. In this paper we sharpen this result, proving that the higher order corrections are analytic with respect to three variables: the first one is just the spacial variable, whereas the second and third (except for n = 2) are irrational powers of it. It is known that this third order term does not appear for the thin-film equation without gravity.
Proceedings of the International Congress of Mathematicians (ICM 2018), 2019
In this paper we consider the problem: ∂tu − ∆u = f (u), u(0) = u0 ∈ exp L p (R N), where p > 1 a... more In this paper we consider the problem: ∂tu − ∆u = f (u), u(0) = u0 ∈ exp L p (R N), where p > 1 and f : R → R having an exponential growth at infinity with f (0) = 0. We prove local well-posedness in exp L p 0 (R N) for f (u) ∼ e |u| q , 0 < q ≤ p, |u| → ∞. However, if for some λ > 0, lim inf s→∞ f (s) e −λs p > 0, then non-existence occurs in exp L p (R N). Under smallness condition on the initial data and for exponential nonlinearity f such that |f (u)| ∼ |u| m as u → 0, N(m−1) 2 ≥ p, we show that the solution is global. In particular, p − 1 > 0 sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on m.
In this paper we consider the initial value problem ∂tu − ∆u = f (u), u(0) = u0 ∈ exp L p (R N), ... more In this paper we consider the initial value problem ∂tu − ∆u = f (u), u(0) = u0 ∈ exp L p (R N), where p > 1 and f : R → R having an exponential growth at infinity with f (0) = 0. Under smallness condition on the initial data and for nonlinearity f such that |f (u)| ∼ e |u| q as |u| → ∞, |f (u)| ∼ |u| m as u → 0, 0 < q ≤ p ≤ m, N(m−1) 2 ≥ p > 1, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on m.
We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation \begin{docume... more We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation \begin{document}$ i\partial_t u +\Delta u = \mu |x|^{-b}|u|^\alpha u,\; u(0)\in H^s({\mathbb R}^N),\; N\geq 1,\; \mu\in {\mathbb C},\; \; b>0 $\end{document} and \begin{document}$ \alpha>0. $\end{document} Only partial results are known for the local existence in the subcritical case \begin{document}$ \alpha<(4-2b)/(N-2s) $\end{document} and much more less in the critical case \begin{document}$ \alpha = (4-2b)/(N-2s). $\end{document} In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for ...
Journal of Mathematical Analysis and Applications, 2020
This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
In this paper, we study global well-posedness and long-time asymptotic behavior of solutions to t... more In this paper, we study global well-posedness and long-time asymptotic behavior of solutions to the nonlinear heat equation with absorption, u t - Δ u + | u | α u = 0 {u_{t}-\Delta u+\lvert u\rvert^{\alpha}u=0} , where u = u ( t , x ) ∈ ℝ {u=u(t,x)\in\mathbb{R}} , ( t , x ) ∈ ( 0 , ∞ ) × ℝ N {(t,x)\in(0,\infty)\times\mathbb{R}^{N}} and α > 0 {\alpha>0} . We focus particularly on highly singular initial values which are antisymmetric with respect to the variables x 1 , x 2 , … , x m {x_{1},x_{2},\ldots,x_{m}} for some m ∈ { 1 , 2 , … , N } {m\in\{1,2,\ldots,N\}} , such as u 0 = ( - 1 ) m ∂ 1 ∂ 2 ⋯ ∂ m | ⋅ | - γ ∈ 𝒮 ′ ( ℝ N ) {u_{0}=(-1)^{m}\partial_{1}\partial_{2}\cdots\partial_{m}\lvert\,{\cdot}\,% \rvert^{-\gamma}\in\mathcal{S}^{\prime}(\mathbb{R}^{N})} , 0 < γ < N {0<\gamma 0 {\alpha>0} . Our approach is to study well-posedness and large time behavior on sectorial domains of the form Ω m = { x ∈ ℝ N : x 1 , … , x m > 0 } {\Omega_{m}=\{x\in\mathbb{R}^{N}:...
Transactions of the American Mathematical Society, 2019
We consider a nonlinear heat equation with a double source: | u | p − 1 u |u|^{p-1}u and | ∇ u | ... more We consider a nonlinear heat equation with a double source: | u | p − 1 u |u|^{p-1}u and | ∇ u | q |\nabla u|^q . This equation has a double interest: in ecology, it was used by Souplet (1996) as a population dynamics model; in mathematics, it was introduced by Chipot and Weissler (1989) as an intermediate equation between the semilinear heat equation and the Hamilton-Jacobi equation. Further interest in this equation comes from its lack of variational structure. In this paper, we intend to see whether the standard blow-up dynamics known for the standard semilinear heat equation (with | u | p − 1 u |u|^{p-1}u as the only source) can be modified by the addition of the second source ( | ∇ u | q |\nabla u|^q ). Here arises a nice critical phenomenon at blow-up: - when q > 2 p / ( p + 1 ) q>2p/(p+1) , the second source is subcritical in size with respect to the first, and we recover the classicial blow-up profile known for the standard semilinear case; - when q = 2 p / ( p + 1 ) q...
In this paper we prove local existence of solutions to the nonlinear heat equation ut = ∆u + a|u|... more In this paper we prove local existence of solutions to the nonlinear heat equation ut = ∆u + a|u| α u, t ∈ (0, T), x = (x1, • • • , xN) ∈ R N , a = ±1, α > 0; with initial value u(0) ∈ L 1 loc R N \ {0} , anti-symmetric with respect to x1, x2, • • • , xm and |u(0)| ≤ C(−1) m ∂1∂2 • • • ∂m(|x| −γ) for x1 > 0, • • • , xm > 0, where C > 0 is a constant, m ∈ {1, 2, • • • , N }, 0 < γ < N and 0 < α < 2/(γ + m). This gives a local existence result with highly singular initial values. As an application, for a = 1, we establish new blowup criteria for 0 < α ≤ 2/(γ + m), including the case m = 0. Moreover, if (N − 4)α < 2, we prove the existence of initial values u0 = λf, for which the resulting solution blows up in finite time Tmax(λf), if λ > 0 is sufficiently small. We also construct blowing up solutions with initial data λnf such that λ [(1 α − γ+m 2) −1 ] n Tmax(λnf) has different finite limits along different sequences λn → 0. Our result extends the known "small lambda" blow up results for new values of α and a new class of initial data.
Proceedings of the American Mathematical Society, 2017
We show the uniqueness of strong solutions for the thin-film equation ut + (uuxxx)x = 0 with init... more We show the uniqueness of strong solutions for the thin-film equation ut + (uuxxx)x = 0 with initial data u(0) = mδ, m > 0 where δ is the Dirac mass at the origin. In particular, the solution is the source type one obtained by Smyth and Hill. The argument is based on an entropy estimate for the equation in self-similar variables.
Nonlinear Analysis: Theory, Methods & Applications, 2017
In this paper we study the nonlinear parabolic equation ∂tu = ∆u + a|x| −γ |u| α u, t > 0, x ∈ R ... more In this paper we study the nonlinear parabolic equation ∂tu = ∆u + a|x| −γ |u| α u, t > 0, x ∈ R N \{0}, N ≥ 1, a ∈ R, α > 0, 0 < γ < min(2, N) and with initial value u(0) = ϕ. We establish local well-posedness in L q (R N) and in C0(R N). In particular, the value q = N α/(2 − γ) plays a critical role. For α > (2 − γ)/N, we show the existence of global self-similar solutions with initial values ϕ(x) = ω(x)|x| −(2−γ)/α , where ω ∈ L ∞ (R N) is homogeneous of degree 0 and ω ∞ is sufficiently small. We then prove that if ϕ(x) ∼ ω(x)|x| −(2−γ)/α for |x| large, then the solution is global and is asymptotic in the L ∞-norm to a self-similar solution of the nonlinear equation. While if ϕ(x) ∼ ω(x)|x| −σ for |x| large with (2 − γ)/α < σ < N, then the solution is global but is asymptotic in the L ∞-norm to e t∆ (ω(x)|x| −σ). The equation with more general potential, ∂tu = ∆u + V (x)|u| α u, V (x)|x| γ ∈ L ∞ (R N), is also studied. In particular, for initial data ϕ(x) ∼ ω(x)|x| −(2−γ)/α , |x| large , we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity.
We study positive blowing-up solutions of systems of the form: ut = δ1∆u + e pv , vt = δ2∆v + e q... more We study positive blowing-up solutions of systems of the form: ut = δ1∆u + e pv , vt = δ2∆v + e qu , with δ1, δ2 > 0 and p, q > 0. We prove single-point blow-up for large classes of radially decreasing solutions. This answers a question left open in a paper of Friedman and Giga (1987), where the result was obtained only for the equidiffusive case δ1 = δ2 and the proof depended crucially on this assumption.
Nous etudions les solutions autosimilaires a symetrie radiale de certaines equations semi-lineair... more Nous etudions les solutions autosimilaires a symetrie radiale de certaines equations semi-lineaires paraboliques avec terme de gradient non lineaire. La these se divise en deux parties independantes. Dans la premiere partie nous demontrons l'existence de solutions autosimilaires definies pour le temps positif, qui sont globales, strictement positives et avec donnees initiales singulieres a l'origine. En particulier, ceci conduit a un resultat de non unicite pour le probleme de Cauchy associe a ces equations dans certains espaces de Lebesgue et de Sobolev. Dans la deuxieme partie, nous considerons le cas unidimensionnel et de plus le gradient est affecte d'une constante negative. Nous demontrons l'existence d'une solution autosimilaire non triviale definie pour le temps negatif. En particulier, ceci implique l'existence d'une solution explosive dont l'ensemble des points d'explosion est reduit a un seul point. Le profil final de cette solution au moment de l'explosion et au voisinage de ce point est different de tous les profils possibles, dans le cas d'un seul point d'explosion, pour cette equation sans terme de gradient, a savoir l'equation de la chaleur avec terme de source
We show that the heat semigroup is well defined on the Banach space $${\mathcal{X}_{m,\gamma} = \... more We show that the heat semigroup is well defined on the Banach space $${\mathcal{X}_{m,\gamma} = \{ \psi:\Omega_m \to \mathbb{R} ;\; |x|^{\gamma +2m}(\prod_{i=1}^m x_i)^{-1}\psi(x) \in L^\infty(\Omega_m)\},}$$Xm,γ={ψ:Ωm→R;|x|γ+2m(∏i=1mxi)-1ψ(x)∈L∞(Ωm)},$${0 < \gamma < N}$$0 0,\, 1\leq i\leq m\},}$$Ωm={(x1,x2,…,xN)∈RN;xi>0,1≤i≤m},$${1\leq m\leq N}$$1≤m≤N. We then investigate the large time behavior of solutions of the heat equation $${u_{t}-\Delta u=0}$$ut-Δu=0 for t > 0 and $${x \in \Omega_m.}$$x∈Ωm. Using certain notions from dynamical systems, we show that the large time behavior is related to the spatial asymptotic behavior of its initial value. Since the space $${\mathcal{X}_{m, \gamma}}$$Xm,γ contains highly singular initial data, which can be extended to all of $${\mathbb{R}^{N}}$$RN by antisymmetry, we also obtain new results on the complexity in the asymptotic behavior of solutions for the heat equation on the whole space.
We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat ... more We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation: ut = D 2α x u ∓ u 2 , t ∈ (0, T), x ∈ R or T, with 0 < α ≤ 1 is well-posed in H s for s ≥ max(−α, 1/2 − 2α) except in the case α = 1/2 where it is shown to be well-posed for s > −1/2 and ill-posed for s = −1/2. As a by-product we improve the known well-posedness results for the heat equation (α = 1) by reaching the end-point Sobolev index s = −1. Finally, in the case 1/2 < α ≤ 1, we also prove optimal results in the Besov spaces B s,q 2 .
We study positive blowing-up solutions of the system: ut − δ∆u = v p , vt − ∆v = u q , as well as... more We study positive blowing-up solutions of the system: ut − δ∆u = v p , vt − ∆v = u q , as well as of some more general systems. For any p, q > 1, we prove single-point blow-up for any radially decreasing, positive and classical solution in a ball. This improves on previously known results in 3 directions: (i) no type I blow-up assumption is made (and it is known that this property may fail); (ii) no equidiffusivity is assumed, i.e. any δ > 0 is allowed; (iii) a large class of nonlinearities F (u, v), G(u, v) can be handled, which need not follow a precise power behavior. As side result, we also obtain lower pointwise estimates for the final blow-up profiles.
Nonlinear Differential Equations and Applications NoDEA, 2015
In this paper we study the existence and the asymptotic behavior of global solutions for a parabo... more In this paper we study the existence and the asymptotic behavior of global solutions for a parabolic system related to the complexvalued heat equation with quadratic nonlinearity: ∂tz = Δz + z 2 , t > 0, x ∈ R N , with initial data z0 = u0 + iv0. We show that if u0(x) ∼ c|x| −2α 1 and v0(x) ∼ c|x| −2α 1 , as |x| → ∞ with α1 ≥ 1, 2α 1 − α1 ≥ 1, N 2α 1 > 1, N 2α 1 > 1 (|c| is sufficiently small), then the solution is global and converges to a self-similar solution. We also establish the existence of four different self-similar behaviors. These behaviors depend on the values of α1 and α 1. In particular, the real and the imaginary parts of the constructed solutions may have different behaviors in the L ∞-norm for large time. Also, the real part may have different behaviors from those known for the real-valued quadratic heat equation.
Transactions of the American Mathematical Society, 2013
In this paper we prove local existence of solutions of the nonlinear heat equation u t = Δ u + | ... more In this paper we prove local existence of solutions of the nonlinear heat equation u t = Δ u + | u | α u , t ∈ ( 0 , T ) , x ∈ R N , u_t = \Delta u + |u|^\alpha u, \; t\in (0,T),\; x\in \mathbb {R}^N, with initial value u ( 0 ) = K ∂ 1 ∂ 2 ⋅ ⋅ ⋅ ∂ m δ , K ≠ 0 , m ∈ { 1 , 2 , ⋯ , N } , 0 > α > 2 / ( N + m ) u(0)=K\partial _{1}\partial _{2}\cdot \cdot \cdot \partial _{m}\delta ,\; K\not =0,\; m\in \{1,\; 2,\; \cdots ,\; N\},\; 0>\alpha >2/(N+m) and δ \delta is the Dirac distribution. In particular, this gives a local existence result with an initial value in a high order negative Sobolev space H s , q ( R N ) H^{s,q}(\mathbb {R}^N) with s ≤ − 2. s\leq -2. As an application, we prove the existence of initial values u 0 = λ f u_0 = \lambda f for which the resulting solution blows up in finite time if λ > 0 \lambda >0 is sufficiently small. Here, f f satisfies in particular f ∈ C 0 ( R N ) ∩ L 1 ( R N ) f\in C_0(\mathbb {R}^N)\cap L^1(\mathbb {R}^N) and is anti-symmetri...
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