We prove the existence of solutions converging to a stationary state for abstract semilinear parabolic problems with a convolution kernel that is unknown (together with the solution). These solutions are suitable perturbations of stationary states. The main tools are maximal regularity results in an $L^1$ (time) setting. The abstract results are applied to a reaction-diffusion integrodifferential system.

Given a bounded open subset $\Omega$ of $\mathbb R^d$ and two positive weight functions $f$ and $g$, the Cheeger sets of $\Omega$ are the subdomains $C$ of finite perimeter of ${{\overline{\Omega}}}$ that maximize the ratio $$ \int_C f(x)\,dx\,\big/\int_{\partial^* C}g(x)\,d{{\mathcal H}}^{d-1} . $$ Existence of Cheeger sets is a well-known fact. Uniqueness is a more delicate issue and is not true in general (although it holds when $\Omega$ is convex and $f\equiv g\equiv 1$ as recently proved in [4]). However, there always exists a unique maximal Cheeger set, and this paper addresses the issue of how to determine this maximal set. We show that in general the approximation by the $p$-Laplacian does not provide, as $p\to 1$, a selection criterion for determining the maximal Cheeger set. On the contrary, a different perturbation scheme, based on the constrained maximization of $\int_\Omega f(u-{\varepsilon}\Phi(u)\big)\,dx$ for a strictly convex function $\Phi$, gives, as ${\varepsilon}\to0$, the desired maximal set.

In this paper we deal with the problem $$\left\{ \begin{array}{rcl} - {\rm div}\, (a(x,u)\nabla u) +{g(x,u,\nabla u)} & = & \lambda h(x)u + f{\mbox{ in }}\Omega,\\ u & = & 0{\mbox{ on }}\partial\Omega. \end{array} \right. $$ The main goal of the work is to get hypotheses on $a$, $g$ and $h$ such that the previous problem has a solution for all $\lambda>0$ and $f\in L^1(\Omega)$. In particular, we focus our attention in the model equation with $a(x,u)= (1+|u|^m)$, $g(x,u,\nabla u)=\frac{m}{2}|u|^{m-2}u|\nabla u|^2$ and $h(x)=\dfrac{1}{|x|^2}$.

In this paper, we are concerned with the existence of $L^\infty$-global bounds for global-in-time solutions of some semilinear parabolic problems. It is well known that every global-in-time solution for the subcritical problem is globally bounded in $L^\infty$, while there exists a global solution which is not bounded in $L^\infty$ globally in time in the critical case. In this paper, we discuss the necessary and sufficient condition for the existence of $L^\infty$-global bounds, which is valid for the subcritical and the critical case in a unified way. Moreover, using our main results, we provide various examples with the critical exponent in which every global-in-time solution has an $L^\infty$-global bound.

In this paper, we establish that the pseudo-differential operator $T_{\sigma}$ associated with the Dunkl operator on the real line is bounded from $L_{p,\alpha}$ into itself for $1 < p < +\infty$, when the symbol $\sigma$ belongs to the class $S^{0}$.

A system of non-linear partial differential equations modeling tumor invasion into surrounding healthy tissue is analyzed. The model incorporates haptotaxis, i.e., the directed migratory response of tumor cells to the extracellular environment, as well as spatial and age structure of the tumor cells. Global existence and uniqueness of non-negative solutions is shown.

This paper concerns the formation of a coincidence set for the positive solution of an equation of the type: $-{\varepsilon} \Delta u=u|a(x)-u|^{\theta} {\operatorname{sgn}} (a(x)-u)$, where ${\varepsilon}$ is a positive parameter, $0 < \theta < 1$ and $a(x)$ is a positive continuous function. Suppose that $\Delta a=0$ in an open subset ${\Omega}_0 \subset {\Omega}$. The positive solution converges to $a(x)$ uniformly on any compact subset of $\Omega$ as ${\varepsilon} \to 0$. It is proved that when ${\varepsilon}$ is sufficiently small, the solution coincides with $a(x)$ somewhere in ${\Omega}_0$ and the coincidence set converges to ${\Omega}_0$ in the Hausdorff distance with the order of $\sqrt{{\varepsilon}}$ as ${\varepsilon} \to 0$. The proof relies on the comparison principle with suitable local comparison functions.