$$\varvec{L^1-L^1}$$ estimates for a doubly dissipative semilinear wave equation

Abstract

In this paper, we derive energy estimates and $$L^1-L^1$$ estimates, for the solution to the Cauchy problem for the doubly dissipative wave equation $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}-\Delta u+u_t-\Delta u_t=0,&{}\quad t\ge 0,\ x\in {\mathbb {R}}^n,\\ (u,u_t)(0,x)=(u_0,u_1)(x). \end{array}\right. } \end{aligned}$$ The solution is influenced both by the diffusion phenomenon created by the damping term $$u_t$$ , and by the smoothing effect brought by the damping term $$-\Delta u_t$$ . Thanks to these two effects, we are able to obtain linear estimates which may be effectively applied to find global solutions in any space dimension $$n\ge 1$$ , to the problems with power nonlinearities $$|u|^p$$ , $$|u_t|^p$$ and $$|\nabla u|^p$$ , in the supercritical cases, by only assuming small data in the energy space, and with $$L^1$$ regularity. We also derive optimal energy estimates and $$L^1-L^1$$ estimates, for the solution to the semilinear problems.