Anzeige
Erschienen in:
2021 | OriginalPaper | Buchkapitel
Sharp Estimate of the Life Span of Solutions to the Heat Equation with a Nonlinear Boundary Condition
verfasst von : Kotaro Hisa
Erschienen in: Geometric Properties for Parabolic and Elliptic PDE's
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
Abstract
Consider the heat equation with a nonlinear boundary condition where N ≥ 1, p > 1, κ > 0 and ψ is a nonnegative measurable function in \({\mathbf {R}}^N_+ :=\{y\in {\mathbf {R}}^N:y_N>0 \}\). Let us denote by T(κψ) the life span of solutions to problem (P). We investigate the relationship between the singularity of ψ at the origin and T(κψ) for sufficiently large κ > 0 and the relationship between the behavior of ψ at the space infinity and T(κψ) for sufficiently small κ > 0. Moreover, we obtain sharp estimates of T(κψ), as κ →∞ or κ → +0.
$$\displaystyle \mathrm {(P)}\qquad \left \{ \begin {array}{ll} \partial _t u=\Delta u,\qquad & x\in {\mathbf {R}}^N_+,\,\,\,t>0,\\ \displaystyle {-\frac {\partial u}{\partial x_N} u}=u^p, & x\in \partial {\mathbf {R}}^N_+,\,\,\,t>0,\\ u(x,0)=\kappa \psi (x),\qquad & x\in \overline {{\mathbf {R}}^N_+}, \end {array} \right . \qquad \qquad $$