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Andrey Shishkov

Publications and source records attributed to Andrey Shishkov.

11 recordsLinked to original sources
arXiv2021

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.

math.AP↗
arXiv2015

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 $\gf(0)=\infty$.

math.AP↗
arXiv2012

Fading absorption in non-linear elliptic equations

We study the equation $-Δu+h(x)|u|^{q-1}u=0$, $q>1$, 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.

math.AP↗
arXiv2011

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-Δu+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.

math.AP↗
arXiv2010

The thin film equation with backwards second order diffusion

In this paper, we focus on the thin film equation with lower order "backwards" diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a "porous media cutoff" of van der Waals forces. We treat in detail the equation $$u_t + \{u^n(u_{xxx} + νu^{m-n}u_x -A u^{M-n} u_x)\}_x=0,$$ where $ν=\pm 1,$ $n>0,$ $M>m,$ and $A \ge 0.$ Global existence of weak nonnegative solutions is proven when $ m-n> -2$ and $A>0$ or $ν=-1,$ and when $-2< m-n<2,$ $A=0,$ $ν=1.$ From the weak solutions, we get strong entropy solutions under the additional constraint that $m-n> -{3}/{2}$ if $ν=1.$ A local energy estimate is obtained when $2 \le n<3 $ under some additional restrictions. Finite speed of propagation is proven when $m>n/2,$ for the case of "strong slippage," $0<n<2,$ when $ν=1$ based on local entropy estimates, and for the case of "weak slippage," $2 \le n<3,$ when $ν=\pm 1$ based on local entropy and energy estimates.

math.AP↗
arXiv2010

Singular solutions to the heat equations with nonlinear absorption and Hardy potentials

We study the existence and nonexistence of singular solutions to the equation $u_t-Δu - \fracκ{|x|^2}u+|x|^αu|u|^{p-1}=0$, $p>1$, in $\R^N\times[0,\infty)$, $N\ge 3$, with a singularity at the point $(0,0)$, that is, nonnegative solutions satisfying $u(x,0)=0$ for $x\ne0$, assuming that $\a>-2$ and $κ<\left(\frac{N-2}2\right)^2$. The problem is transferred to the one for a weighted Laplace-Beltrami operator with a non-linear absorbtion, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a {\it source solution} with a multiple of the Dirac mass as initial datum, or as a unique {\it very singular solution}, leads to a complete classification of singular solutions to the original problem, which exist if and only if $p<1+\frac{2(2+α)}{N+2+\sqrt{(N-2)^2-4κ}}$.

math.AP↗
arXiv2009

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy-Dirichlet problem to the semilinear $2m$-order ($m \geq 1$) parabolic equation $u_t+Lu+a(x) |u|^{q-1}u=0$, $0 2m$ and $\displaystyle \int_0^1 s^{-1} \text{meas} \{x \in Ω: |a(x)| \leq s \}^\frac{2m}{N} ds < + \infty$, then the solution $u$ vanishes in a finite time. When $N=2m$, the condition becomes $\displaystyle \int_0^1 s^{-1} (\text{meas} \{x \in Ω: |a(x)| \leq s \}) (-\ln \text{meas} \{x \in Ω: |a(x)| \leq s \}) ds < + \infty$.

math.AP↗
arXiv2008

The balance between diffusion and absorption in semilinear parabolic equations

Let $h:[0,\infty)\mapsto [0,\infty)$ be continuous and nondecreasing, $h(t)>0$ if $t>0$, and $m,q$ be positive real numbers. We investigate the behavior when $k\to\infty$ of the fundamental solutions $u=u_{k}$ of $\prt_{t} u-Δu^m+h(t)u^q=0$ in $Ω\ti (0,T)$ satisfying $u_{k}(x,0)=kδ_0$. The main question is wether the limit is still a solution of the above equation with an isolated singularity at $(0,0)$, or a solution of the associated ordinary differential equation $ u'+h(t)u^q=0$ which blows-up at $t=0$.

math.AP↗
arXiv2008

Diffusion versus absorption in semilinear parabolic equations

We study the limit, when $k\to\infty$, of the solutions $u=u_{k}$ of (E) $\prt_{t}u-Δu+ h(t)u^q=0$ in $\BBR^N\ti (0,\infty)$, $u_{k}(.,0)=kδ_{0}$, with $q>1$, $h(t)>0$. If $h(t)=e^{-\gw(t)/t}$ where $\gw>0$ satisfies to $\int_{0}^1\sqrt{\gw(t)}t^{-1}dt<\infty$, the limit function $u_{\infty}$ is a solution of (E) with a single singularity at $(0,0)$, while if $\gw(t)\equiv 1$, $u_{\infty}$ is the maximal solution of (E). We examine similar questions for equations such as $\prt_{t}u-\Gd u^m+ h(t)u^q=0$ with $m>1$ and $\prt_{t}u-\Gd u+ h(t)e^{u}=0$.

math.AP↗
arXiv2008

Long-time extinction of solutions of some semilinear parabolic equations

We study the long time behaviour of solutions of semi-linear parabolic equation of the following type $\partial_t u-Δu+a_0(x)u^q=0$ where $a_0(x) \geq d_0 \exp(\frac{ω(|x|)}{|x|^2})$, $d_0>0$, $1>q>0$ and $ω$ a positive continuous radial function. We give a Dini-like condition on the function $ω$ by two different method which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.

math.AP↗