A note on maximal solutions of nonlinear parabolic equations with absorption

If $\Omega$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-\Delta u+f(u)=0$ in $Q_\infty^\Omega:=\Omega\times (0,\infty)$, $u=\infty$ on the parabolic boundary $\partial_{p}Q$. We prove that in most cases, the existence and uniqueness is reduced to the same property for the associated stationary equation in $\Omega$.Comment: \`A para\^itre \`a Asymptotic Analysi

Existence and stability of solutions of general semilinear elliptic equations with measure data

We study existence and stability for solutions of $Lu+g(x; u) = \omega$ in the closure of open set $\Omega$ where L is a second order elliptic operator, $g$ a Caratheodory function and $\omega$ a measure in $\bar\Omega$. We present a uni ed theory of the Dirichlet problem and the Poisson equation. We prove the stability of the problem with respect to weak convergence of the data.Comment: to appear in Advanced Nonlinear Studie

Boundary Value Problems with Measures for Elliptic Equations with Singular Potentials

We study the boundary value problem with Radon measures for nonnegative solutions of $-\Delta u+Vu=0$ in a bounded smooth domain \Gw, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give sufficient conditions on a Radon measure \gm on \prt\Gw so that the problem can be solved. We study the reduced measure associated to this equation as well as the boundary trace of positive solutions

Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions of (E) $(-\Delta)^\alpha u+g(u)=\nu$ in a bounded regular domain $\Omega$ in $\R^N (N\ge2)$ which vanish on $\R^N\setminus\Omega$, where $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ is a Radon measure and $g$ is a nondecreasing function satisfying some extra hypothesis. When $g$ satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where $\nu$ is Dirac measure, we characterize the asymptotic behavior of the solution. When $g(r)=|r|^{k-1}r$ with $k$ supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved

Propagation of Singularities of Nonlinear Heat Flow in Fissured Media

In this paper we investigate the propagation of singularities in a nonlinear parabolic equation with strong absorption when the absorption potential is strongly degenerate following some curve in the $(x,t)$ space. As a very simplified model, we assume that the heat conduction is constant but the absorption of the media depends stronly of the characteristic of the media. More precisely we suppose that the temperature $u$ is governed by the following equation \label{I-1} \partial_{t}u-\Delta u+h(x,t)u^p=0\quad \text{in}Q_{T}:=R^N\times (0,T) where $p>1$ and $h\in C(\bar Q_{T})$. We suppose that $h(x,t)>0$ except when $(x,t)$ belongs to some space-time curve.Comment: To appear in Comm. Pure Appl. Ana

Separable solutions of some quasilinear equations with source reaction

We study the existence of singular separable solutions to a class of quasilinear equations with reaction term. In the 2-dim case, we use a dynamical system approach to construct our solutions.Comment: 34 page

Boundary trace of positive solutions of supercritical semilinear elliptic equations in dihedral domains

We study the generalized boundary value problem for (E)\; $-\Delta u+|u|^{q-1}u=0$ in a dihedral domain \Gw, when $q>1$ is supercritical. The value of the critical exponent can take only a finite number of values depending on the geometry of \Gw. When \gm is a bounded Borel measure in a k-wedge, we give necessary and sufficient conditions in order it be the boundary value of a solution of (E). We also give conditions which ensure that a boundary compact subset is removable. These conditions are expressed in terms of Bessel capacities $B_{s,q'}$ in \BBR^{N-k} where $s$ depends on the characteristics of the wedge. This allows us to describe the boundary trace of a positive solution of (E)Comment: To appear Ann. Sc.Norm. Sup. Pisa Cl. Sci. arXiv admin note: substantial text overlap with arXiv:0907.100

Capacitary estimates of solutions of semilinear parabolic equations

We prove an almoste representation formula for positive solutions of semilinear heat equations with power-type absorption the initial trace of which is the indicatrix function of a compact set. The representations involves a Wiener test via Bessel capacities.Comment: 46 page

Initial value problems for diffusion equations with singular potential

Let $V$ be a nonnegative locally bounded function defined in Q_\infty:=\BBR^n\times(0,\infty). We study under what conditions on $V$ and on a Radon measure \gm in $\mathbb{R}^d$ does it exist a function which satisfies \partial_t u-\xD u+ Vu=0 in $Q_\infty$ and u(.,0)=\xm. We prove the existence of a subcritical case in which any measure is admissible and a supercritical case where capacitary conditions are needed. We obtain a general representation theorem of positive solutions when $t V(x,t)$ is bounded and we prove the existence of an initial trace in the class of outer regular Borel measures.Comment: To appear in Contemporary Mathematic