Bounds for the extremal parameter of nonlinear eigenvalue problems and application to the explosion problem in a flow

TL;DR

This paper establishes bounds for the extremal parameter in nonlinear eigenvalue problems and explores how flow properties influence the explosion behavior in a flow, with applications to specific operators and flow configurations.

Contribution

It provides new bounds for the extremal parameter and solutions, and characterizes the effect of flow divergence properties on the explosion problem in bounded domains.

Findings

Bounds for extremal parameter and solution established.

Flow divergence properties determine the behavior of solutions as flow intensity increases.

Extremal parameter behavior is fully characterized for certain flow types.

Abstract

We consider the nonlinear eigenvalue problem $ L u = \lambda f(u) $, posed in a smooth bounded domain $ \Omega \subseteq \Bbb{R}^{N} $ with Dirichlet boundary condition, where $ L $ is a uniformly elliptic second-order linear differential operator, $ \lambda > 0 $ and $ f:[0,a_{f}) \rightarrow \Bbb{R}_{+} $ $ (0 < a_{f} \leqslant \infty)$ is a smooth, increasing and convex nonlinearity such that $ f(0) > 0 $ and which blows up at $ a_{f} $. First we present some upper and lower bounds for the extremal parameter $ \lambda^{*} $ and the extremal solution $ u^{*} $. Then we apply the results to the operator $ L_A = - \Delta + A c(x) $ with $ A>0 $ and $ c(x) $ is a divergence-free flow in $ \Omega $. We show that, if $\psi_{A,\Omega}$ is the maximum of the solution $\psi_{A}(x)$ of the equation $ L_A u = 1 $ in $\Omega$ with Dirichlet boundary condition, then for any incompressible flow $ c(x) $ we have, $\psi_{A,\Omega} \longrightarrow 0$ as $A \longrightarrow \infty$ if and only if $c(x)$ has no non-zero first integrals in $H_{0}^{1}(\Omega)$. Also, taking $ c(x)=-x\rho(|x|) $ where $\rho$ is a smooth real function on $[0,1]$ then $c(x)$ is never divergence-free in unit ball $ B\subset \Bbb{R}^{N} $, but our results completely determine the behaviour of the extremal parameter $ \lambda^{*}_{A} $ as $ A \longrightarrow \infty $.