Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems
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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 ≤ pc. 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.
CitationKuiper, H. J. (2003). Life span of nonnegative solutions to certain quasilinear parabolic Cauchy problems. Electronic Journal of Differential Equations, 2003(66), pp. 1-11.
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