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 $u_{tt}-\Delta u=u_t|u_t|^{p-1}$.
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.
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 $u_{tt}-\Delta u=u_t|u_t|^{p-1}$.
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