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 prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\p... more In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ \Delta^2 u=f(u),\;t>0,\;x\in\R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $f(u)\sim u^m$ as $u\to 0,$ $m$ integer and $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain a 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.
This paper is devoted to system of semilinear heat equations with exponential-growth nonlinearity... more This paper is devoted to system of semilinear heat equations with exponential-growth nonlinearity in two-dimensional space which is the analogue of the scalar model problem studied in [S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Local well posedness of a 2D semilinear heat equation, Bull. Belg. Math. Soc.21 (2014) 1–17]. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space (H1 × H1)(ℝ2). The uniqueness part is nontrivial although it follows Brezis–Cazenave's proof [H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math.68 (1996) 73–90] in the case of monomial nonlinearity in dimension d ≥ 3. Next, we show that in the defocusing case our solution is bounded, and therefore exists globally in time. Finally, for this system, we treat the question of blow-up in finite time under the negativity condition on the energy functional. The technique to be used is adapted from [Bull. Belg. Math. Soc. 2...
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.
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 prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\p... more In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ \Delta^2 u=f(u),\;t>0,\;x\in\R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $f(u)\sim u^m$ as $u\to 0,$ $m$ integer and $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain a 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.
This paper is devoted to system of semilinear heat equations with exponential-growth nonlinearity... more This paper is devoted to system of semilinear heat equations with exponential-growth nonlinearity in two-dimensional space which is the analogue of the scalar model problem studied in [S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Local well posedness of a 2D semilinear heat equation, Bull. Belg. Math. Soc.21 (2014) 1–17]. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space (H1 × H1)(ℝ2). The uniqueness part is nontrivial although it follows Brezis–Cazenave's proof [H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math.68 (1996) 73–90] in the case of monomial nonlinearity in dimension d ≥ 3. Next, we show that in the defocusing case our solution is bounded, and therefore exists globally in time. Finally, for this system, we treat the question of blow-up in finite time under the negativity condition on the energy functional. The technique to be used is adapted from [Bull. Belg. Math. Soc. 2...
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.
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