Sharp upper bound for lifespan of solutions to some critical semilinear parabolic, dispersive and hyperbolic equations via a test function method

This paper is concerned with the blowup phenomena for initial-boundary value problem { tau partial derivative(2)(t)u(x, t) - Delta u(x, t) + a(x)partial derivative(t)u(x, t) = lambda vertical bar u(x, t)vertical bar(p) , (x, t) is an element of C-Sigma x (0, T), u(x, t) = 0, (x, t) is an element of partial derivative C-Sigma x (0, T), (0.1) u(x, 0) = epsilon f (x), x is an element of C-Sigma, tau partial derivative(t)u(x, 0) = tau epsilon g (x), x is an element of C-Sigma, where C-Sigma is a cone-like domain in R-N (N >= 2) defined as C-Sigma = int {r omega is an element of R-N; r >= 0, omega is an element of Sigma} with a connected open set Sigma in SN-1 with smooth boundary partial derivative Sigma If N = 1, then we only consider two cases C-Sigma = (0, infinity) and C-Sigma = R. Here a(x) is a non-zero coefficient of partial derivative(t)u which could be complex-valued and space-dependent, lambda is an element of C is a fixed constant, and epsilon > 0 is a small parameter. The constants tau = 0, 1 switch the parabolicity and hyperbolicity of the problem (0.1). The result proposes an argument for sharp upper lifespan estimates of solutions to (0.1) by a test function method based on Mitidieri and Pokhozhaev (2001). The crucial idea is to introduce an ordinary differential inequality with a parameter as a variable. (C) 2018 Elsevier Ltd. All rights reserved.