Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems
Date
2003-06-13
Authors
Kuiper, Hendrik J.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We consider the problem
ρ(x)ut - ∆um = h(x, t)u1+p, x ∈ ℝN, t > 0,
with nonnegative, nontrivial, continuous initial condition,
u(x, 0) = u0(x) ≢ 0, u0(x) ≥ 0, x ∈ ℝN.
An integral inequality is obtained that can be used to find an exponent pc such that this problem has no nontrivial global solution when p ≤ p c. This integral inequality may also be used to estimate the maximal T > 0 such that there is a solution for 0 ≤ t < T. This is illustrated for the case ρ ≡ 1 and h ≡ 1 with initial condition u(x, 0) = σu0(x), σ > 0, by obtaining a bound of the form T ≤ C0σ-ϧ.
Description
Keywords
Nonlinear parabolic equation, Blow-up, Lifespan, Critical exponent
Citation
Kuiper, H. J. (2003). Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems. <i>Electronic Journal of Differential Equations, 2003</i>(66), pp. 1-11.
Rights
Attribution 4.0 International