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 $u_t+Lu+a(x) |u|^{q-1}u=0$, $0<q<1$ with $a(x) \geq 0$ bounded in the bounded domain $\Omega \subset \R^N$. We prove that if $N>2m$ and $\displaystyle \int_0^1 s^{-1} \text{meas} \{x \in \Omega : |a(x)| \leq s \}^\frac{2m}{N} ds < + \infty$, then the solution $u$ vanishes in a finite time. When $N=2m$, the condition becomes $\displaystyle \int_0^1 s^{-1} (\text{meas} \{x \in \Omega : |a(x)| \leq s \}) (-\ln \text{meas} \{x \in \Omega : |a(x)| \leq s \}) ds < + \infty$

Asymptotic Estimates for a Variational Problem Involving a Quasilinear Operator in the Semi-Classical Limit

Let Ω be a domain of ℝN. We study the infimum λ1(h) of the functional ∫Ω|∇u|p+h −p V(x)|u|p dx in W 1,p(Ω) for ||u|| LP(Ω)= 1 where h > 0 tends to zero and V is a measurable function on Ω. When V is bounded, we describe the behaviour of λ1(h), in particular when V is radial and 'slowly' decaying to zero. We also study the limit of λ1(h) when h→ 0 for more general potentials and show a necessary and sufficient condition for λ1(h) to be bounded. A link with the tunelling effect is established. We end with a theorem of existence for a first eigenfunction related to λ1(h)

Semi-classical analysis and vanishing properties of solutions to quasilinear equations

Let $Omega$ be an open bounded subset of $mathbb{R}^N$ and $b$ a measurable nonnegative function in $Omega$. We deal with the time compact support property for $$u_t - Delta u + b(x)|u|^{q-1} u = 0$$ for $p geq 2$ and $$u_t - mathop{m div} ( |abla u|^{p-2} abla u ) + b(x)|u|^{q-1} u = 0$$ with $m geq 1$ where $0 leq q <1$. We give criteria associated to the first eigenvalue of some quasilinear Schr"odinger operators in semi-classical limits. We also provide a lower bound for this eigenvalue

Semi-classical analysis and vanishing properties of solutions to quasilinear equations

Let $Omega$ be an open bounded subset of $mathbb{R}^N$ and $b$ a measurable nonnegative function in $Omega$. We deal with the time compact support property for $$u_t - Delta u + b(x)|u|^{q-1} u = 0$$ for $p geq 2$ and $$u_t - mathop{m div} ( |abla u|^{p-2} abla u ) + b(x)|u|^{q-1} u = 0$$ with $m geq 1$ where $0 leq q <1$. We give criteria associated to the first eigenvalue of some quasilinear Schr"odinger operators in semi-classical limits. We also provide a lower bound for this eigenvalue

Long-time extinction of solutions of some semilinear parabolic equations

International audienceWe study the long-time behavior of solutions of semilinear parabolic equation of the following type ∂t u − ∆u + a0(x)u^q = 0 where a0(x) ≥ d0 exp(− [ω(|x|)]/|x|2 ), d0 > 0, 1 > q > 0, and ω is a positive continuousradial function. We give a Dini-like condition on the function ω by two different methods 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

Long-time extinction of solutions of some semilinear parabolic equations

International audienceWe study the long time behaviour of solutions of semi-linear parabolic equation of the following type $\partial_t u-\Delta u+a_0(x)u^q=0$ where $a_0(x) \geq d_0 \exp\left( \frac{\omega(|x|)}{|x|^2}\right)$, $d_0>0$, $1>q>0$ and $\omega$ a positive continuous radial function. We give a Dini-like condition on the function $\omega$ 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

Shishkov A.: Long-time extinction of solutions of some semilinear parabolic equations

parabolic equation

Abstract results on the finite extinction time property: application to a singular parabolic equation

We start by studying the finite extinction time for solutions of the abstract Cauchy problem u(t) + Au + Bu = 0 where A is a maximal monotone operator and B is a positive operator on a Hilbert space H. We use a suitable spectral energy method to get some sufficient conditions which guarantee this property. As application we consider a singular semilinear parabolic equation: Au = -Delta u, Bu = a(x)u(q), a(x) >= 0 bounded and -1 < q < 1, on a regular bounded domain Omega and Dirichlet boundary conditions