Global Solvability and L^p Decay for the Semilinear Dissipative Wave Equations in Four and Five Dimensions

Abstract

We study the global existence, uniqueness, and asymptotic behavior of solutions to the Cauchy problem for the semilinear dissipative wave equations: (∅+∂_t)u =|u|^α+1 in R^N × (0, ∞) with u|_t=0=εu₀ and ∂_tu|_t=0 = εu₁ for a small parameter ε > 0. Here, we do not assume any compactly support conditions on the initial data (u₀, u₁). When dimension N = 4, 5 and α is greater than a critical number 2/N which is often called Fujita's exponent, we solve the global in time solvability problem and we derive the sharp decay rates of L^p norm with p ≥ 1 of the solutions.