Blow-up Time of Solutions for a Fast Diffusion Equation with a Variable Exponent

抄録

This paper is concerned with the Cauchy problem for a fast diffusion equation involving a variable exponent u_t = Δu^m + u^p(x) in Rⁿ, where m is a constant such that max{0, 1 − 2/n} < m < 1 and p(x) is a continuous bounded function such that 1 < p⁻ = inf p ≤ p(x) ≤ sup p = p⁺. Since the thermal conductively mu^m−1 ↑ ∞ when u ↓ 0, mathematically u_t = Δu^m + u^p(x) represents a fast diffusion with source. The initial condition u₀(x) is assumed to be continuous, nonnegative and bounded. For the non-decaying initial data at space infinity, any nontrivial nonnegative solutions blow up in finite time. We give the upper bound of the blow-up time of positive solutions of a fast diffusion equation for the non-decaying initial data at space infinity.