Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of a class of solutions of nonlinear parabolic equations with a degenerate strong absorption. We prove that two types of phenomena can occur: the pointwise singularity or the formation of razor blades (or persistent singularities).Comment: 29 page

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

Admissible initial growth for diffusion equations with weakly superlinear absorption

We study the admissible growth at infinity of initial data of positive solutions of \prt\_t u-\Gd u+f(u)=0 in \BBR\_+\ti\BBR^N when $f(u)$ is a continuous function, {\it mildly} superlinear at infinity, the model case being f(u)=u\ln^\ga (1+u) with 1\textless{}\ga\textless{}2. We prove in particular that if the growth of the initial data at infinity is too strong, there is no more diffusion and the corresponding solution satisfies the ODE problem \prt\_t \gf+f(\gf)=0 on \BBR\_+ with \gf(0)=\infty.Comment: Communications in Contemporary Mathematics, to appea

Large and very singular solutions to semilinear elliptic equations

We consider equation $-\Delta u+f(x,u)=0$ in smooth bounded domain $\Omega\in\mathbb{R}^N$, $N\geqslant2$, with $f(x,r)>0$ in $\Omega\times\mathbb{R}^1_+$ and $f(x,r)=0$ on $\partial\Omega$. We find the condition on the order of degeneracy of $f(x,r)$ near $\partial\Omega$, which is a criterion of the existence-nonexistence of a very singular solution with a strong point singularity on $\partial\Omega$. Moreover, we prove that the mentioned condition is a sufficient condition for the uniqueness of a large solution and conjecture that this condition is also a necessary condition of the uniqueness.Comment: 29 page

Diffusion versus absorption in semilinear elliptic equations

International audienceWe study the limit behaviour of a sequence of singular solutions of a nonlinear elliptic equation with a strongly degenerate absorption term at the boundary of the domain. We give sharp conditions on the level of degeneracy in order the pointwise singularity not to propagate along the boundary

Diffusion versus absorption in semilinear parabolic equations

We study the limit, when $k\to\infty$, of the solutions $u=u_{k}$ of (E) \prt_{t}u-\Delta u+ h(t)u^q=0 in \BBR^N\ti (0,\infty), $u_{k}(.,0)=k\delta_{0}$, with $q>1$, $h(t)>0$. If h(t)=e^{-\gw(t)/t} where \gw>0 satisfies to \int_{0}^1\sqrt{\gw(t)}t^{-1}dt1 and \prt_{t}u-\Gd u+ h(t)e^{u}=0

Fading absorption in non-linear elliptic equations

We study the equation $-\Delta u+h(x)|u|^{q-1}u=0$, $q>1$, in R^N_+=R^{N-1}\ti R_+ where $h\in C(\bar{R^N_+})$, $h\geq 0$. Let $(x_1,..., x_N)$ be a coordinate system such that $R^N_+=[x_N>0]$ and denote a point x\in \RN by $(x',x_N)$. Assume that $h(x', x_N)>0$ when $x'\neq 0$ but $h(x',x_N)\to 0$ as $|x'|\to 0$. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior