Related papers: Fractional semilinear heat equations with singular…

This article is concerned with a semilinear time-fractional diffusion equation with a superlinear convex semilinear term in a bounded domain $\Omega$ with the homogeneous Dirichlet, Neumann, Robin boundary conditions and non-negative and…

We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then, additional existence and continuation theorems are obtained, ensuring global existence of solutions.

We give a simple proof of a lower bound for the Dirichlet heat kernel in terms of the Gaussian heat kernel. Using this we establish a non-existence result for semilinear heat equations with zero Dirichlet boundary conditions and initial…

We consider nonnegative solutions of the quasilinear heat equation $\partial_t u = \tfrac{1}{2} u \partial_x^2 u$ in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are…

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

Applying the method of moving planes in integral forms, we establish radial symmetry for positive solutions to a class of semilinear equations involving the fractional Laplacian in the unit ball and obtain Liouville type theorems concerning…

We investigate the large-time behavior of the sign-changing solution of the inhomogeneous semilinear heat equation with a forcing term depending of time and space. we identify the critical exponent for this problem, which separates the…

The class of problems treated here are elliptic partial differential equations with a homogeneous boundary condition and a non-linear perturbation obtained by composition with a fixed smooth function. The existence of solutions is obtained…

In this note we analyse \emph{quantitative} approximation properties of a certain class of \emph{nonlocal} equations: Viewing the fractional heat equation as a model problem, which involves both \emph{local} and \emph{nonlocal}…

In this paper, we study the Cauchy problem for the linear plate equation with mass term and its applications to semilinear models. For the linear problem we obtain $L^p-L^q$ estimates for the solutions in the full range $1\leq p\leq q\leq…

We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We…

We investigate finite-time blow-up for nonnegative solutions to the Cauchy problem associated with semilinear parabolic equations driven by a mixed local--nonlocal operator. The reaction term is assumed to satisfy suitable structural…

In this paper, we consider the regularity of weak solutions (in an appropriate space) to the elliptic partial differential equation \begin{equation*} (-\Delta_{p})^{s} u + (-\Delta_{q})^{s} u = f(x) \quad \text{in} \quad \mathbb{R}^{N},…

We investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population…

We consider a blow-up solution for a strongly perturbed semilinear heat equation with Sobolev subcritical power nonlinearity. Working in the framework of similarity variables, we find a Lyapunov functional for the problem. Using this…

We produce a finite time blow-up solution for nonlinear fractional heat equation ($\partial_t u + (-\Delta)^{\beta/2}u=u^k$) in modulation and Fourier amalgam spaces on the torus $\mathbb T^d$ and the Euclidean space $\mathbb R^d.$ This…

By developing new techniques we establish local existence and uniqueness theorems for an initial value problem involving a nonlinear equation in the sense of Riemann-Liouville fractional derivative in the case that the nonlinear function on…

We discuss exponential decay in $L^p(R^N)$, $1\leq p \leq \infty$, of solutions of a fractional Schr\"odinger parabolic equation with a locally uniformly integrable potential. The exponential type of the semigroup of solutions is considered…

We shed light on a long-standing open question for the semilinear heat equation $u_t = \Delta u + |u|^{p-1} u$. Namely, without any restriction on the exponent $p>1$ nor on the smooth domain~$\Omega$, we prove that the critical $L^q$ norm…

We study existence, uniqueness, norm estimates and asymptotic time behaviour (in some cases can be claimed to be sharp) for the solution of a general evolutionary integral (differential) equation of scalar type on a locally compact…