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We study the local and global existence of solutions to a semilinear evolution equation driven by a mixed local-nonlocal operator of the form \( L = -\Delta + (-\Delta)^{\alpha/2} \), where \( 0 < \alpha < 2 \). The Cauchy problem under…

We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$,…

We study pointwise behavior of positive solutions to nonlinear integral equations, and related inequalities, of the type \begin{equation*} u(x) - \int_\Omega G(x, y) \, g(u(y)) d \sigma (y) = h, \end{equation*} where $(\Omega, \sigma)$ is a…

We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at…

The aim of this paper is to study the long-time dynamics of solutions of the evolution system \[ \begin{cases} u_{tt} - \Delta u + u + \eta(-\Delta)^{\frac{1}{2}}u_t + a_{\epsilon}(t)(-\Delta)^{\frac{1}{2}}v_t = f(u), & \; (x, t) \in \Omega…

This paper is concerned with a predator-prey model in $N$-dimensional spaces ($N=1, 2, 3$), given by \begin{align*}\left\{\begin{aligned} &\frac{\partial u}{\partial t}=\Delta u-\chi\nabla\cdot(u\nabla v),\\ &\frac{\partial v}{\partial…

We consider the Cauchy problem for the nonstationary discrete p-Laplacian with inhomogeneous density \r{ho}(x) on an infinite graph which supports the Sobolev inequality. For nonnegative solutions when p > 2, we prove the precise rate of…

In this work we consider the non local evolution problem \[ \begin{cases} \partial_t u(x,t)=-u(x,t)+g(\beta K(f\circ u)(x,t)+\beta h), ~x \in\Omega, ~t\in[0,\infty[;\\ u(x,t)=0, ~x\in\mathbb{R}^N\setminus\Omega, ~t\in[0,\infty[;\\…

This paper deals with the following Cauchy problem to nonlinear time fractional non-autonomous integro-differential evolution equation of mixed type via measure of noncompactness $$ \left\{\begin{array}{ll} ^CD^{\alpha}_tu(t)+A(t)u(t)=…

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $D^{\alpha}_Cu(t)=Au(t)+f(t)$ on the half line, where $D^{\alpha}_Cu(t)$ is the derivative of the function $u$ in Caputo's sense,…

This paper studies the properties of solutions for a double nonlinear time-dependent parabolic equation with variable density, not in divergence form with a source or absorption. The problem is formulated as a partial differential equation…

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…

We study the asymptotic behavior of solutions of the Cauchy problem associated to a quantitative genetics model with a sexual mode of reproduction. It combines trait-dependent mortality and a nonlinear integral reproduction operator "the…

We study the asymptotic behavior for nonlocal diffusion equations $\partial_tu=\mathcal{J}u-\chi_0u$ in $\mathbb{R}^n\times(0,\infty)$ and obtain a sufficient condition so that solutions of the Cauchy problem decay in time at the rate of a…

We study the Cauchy problem for the quasilinear wave equation $ \partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x $ with $a \geq 0$ and show a result for the local in time existence under new conditions. In the previous results, it is…

The Cauchy problem in $\mathbb R^n$ is considered for \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v),\\ 0 = \Delta v + u. \end{array} \right. \end{eqnarray*} For each $n\ge 10$, a statement on stability…

We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the…

We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $\Omega$, $u=0$ in $\R^n\setminus\Omega$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,d\rho \int_{S^{n-1}}\,da(\omega)\, \frac{…

This paper studies the large time behavior of solutions to semi-linear Cauchy problems with quadratic nonlinearity in gradients. The Cauchy problem considered has a general state space and may degenerate on the boundary of the state space.…

We deal with existence, uniqueness, and regularity for solutions of the boundary value problem $$ \begin{cases} {\mathcal L}^s u = \mu &\quad \text{in $\Omega$}, u(x)=0 \quad &\text{on} \ \ \mathbb{R}^N\backslash\Omega, \end{cases} $$ where…