In this paper, we investigate the local existence and the finite-time blow-up of solutions of sem... more In this paper, we investigate the local existence and the finite-time blow-up of solutions of semilinear parabolic system with nonlocal in time nonlinearity. In addition, we also give the blow-up rate and necessary conditions for local and global existence.
We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equatio... more We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\rightarrow\infty$ of small data solutions have been established in the case when $1\leq n\leq3.$ Moreover, we derive a blow-up result under some positive data in any dimensional space.
We prove local existence and uniqueness of the solution $(u,u_t)\in C^0([0,T];H^1\times L^2(\math... more We prove local existence and uniqueness of the solution $(u,u_t)\in C^0([0,T];H^1\times L^2(\mathbb{R}^N))$ of the semilinear wave equation $u_{tt}-\Delta u=u_t|u_t|^{p-1}$.
Journal of Mathematical Analysis and Applications, 2011
We study local existence and uniqueness in the phase space H µ × H µ−1 (R N ) of the solution of ... more We study local existence and uniqueness in the phase space H µ × H µ−1 (R N ) of the solution of the semilinear wave equation u tt − ∆u = u t |u t | p−1 for p > 1.
In this paper, we investigate the local existence and the finite-time blow-up of solutions of sem... more In this paper, we investigate the local existence and the finite-time blow-up of solutions of semilinear parabolic system with nonlocal in time nonlinearity. In addition, we also give the blow-up rate and necessary conditions for local and global existence.
We consider the Cauchy problem in R n , n ≥ 1, for a semilinear damped wave equation with nonline... more We consider the Cauchy problem in R n , n ≥ 1, for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as t → ∞ of small data solutions have been established in the case when 1 ≤ n ≤ 3. Moreover, we derive a blow-up result under some positive data for in any dimensional space. It turns out that the critical exponent indeed coincides with the one to the corresponding semilinear heat equation.
In this article, we study the local existence of solutions for a wave equation with a nonlocal in... more In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.
We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$... more We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$ This model describes the evolution of the position of each atom in a crystal, and is mathematically given by an infinite system of coupled first order ODEs. We prove that for a suitable rescaling of this model, the solution converges to the solution of a Peierls-Nabarro model, which is a coupled system of two PDEs (typically an elliptic PDE in a domain with an evolution PDE on the boundary of the domain). This passage from the discrete model to a continuous model is done in the framework of viscosity solutions.
The large time behavior of non-negative solutions to the reaction–diffusion equation \({\partial_... more The large time behavior of non-negative solutions to the reaction–diffusion equation \({\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}\) , \({(\alpha\in(0,2], \;p > 1)}\) posed on \({\mathbb{R}^N}\) and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.
In this paper, we investigate the local existence and the finite-time blow-up of solutions of sem... more In this paper, we investigate the local existence and the finite-time blow-up of solutions of semilinear parabolic system with nonlocal in time nonlinearity. In addition, we also give the blow-up rate and necessary conditions for local and global existence.
We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equatio... more We consider the Cauchy problem in $\mathbb{R}^n,$ $n\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\rightarrow\infty$ of small data solutions have been established in the case when $1\leq n\leq3.$ Moreover, we derive a blow-up result under some positive data in any dimensional space.
We prove local existence and uniqueness of the solution $(u,u_t)\in C^0([0,T];H^1\times L^2(\math... more We prove local existence and uniqueness of the solution $(u,u_t)\in C^0([0,T];H^1\times L^2(\mathbb{R}^N))$ of the semilinear wave equation $u_{tt}-\Delta u=u_t|u_t|^{p-1}$.
Journal of Mathematical Analysis and Applications, 2011
We study local existence and uniqueness in the phase space H µ × H µ−1 (R N ) of the solution of ... more We study local existence and uniqueness in the phase space H µ × H µ−1 (R N ) of the solution of the semilinear wave equation u tt − ∆u = u t |u t | p−1 for p > 1.
In this paper, we investigate the local existence and the finite-time blow-up of solutions of sem... more In this paper, we investigate the local existence and the finite-time blow-up of solutions of semilinear parabolic system with nonlocal in time nonlinearity. In addition, we also give the blow-up rate and necessary conditions for local and global existence.
We consider the Cauchy problem in R n , n ≥ 1, for a semilinear damped wave equation with nonline... more We consider the Cauchy problem in R n , n ≥ 1, for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as t → ∞ of small data solutions have been established in the case when 1 ≤ n ≤ 3. Moreover, we derive a blow-up result under some positive data for in any dimensional space. It turns out that the critical exponent indeed coincides with the one to the corresponding semilinear heat equation.
In this article, we study the local existence of solutions for a wave equation with a nonlocal in... more In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.
We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$... more We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$ This model describes the evolution of the position of each atom in a crystal, and is mathematically given by an infinite system of coupled first order ODEs. We prove that for a suitable rescaling of this model, the solution converges to the solution of a Peierls-Nabarro model, which is a coupled system of two PDEs (typically an elliptic PDE in a domain with an evolution PDE on the boundary of the domain). This passage from the discrete model to a continuous model is done in the framework of viscosity solutions.
The large time behavior of non-negative solutions to the reaction–diffusion equation \({\partial_... more The large time behavior of non-negative solutions to the reaction–diffusion equation \({\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}\) , \({(\alpha\in(0,2], \;p > 1)}\) posed on \({\mathbb{R}^N}\) and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.
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Papers by Ahmad Fino