$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

Abstract

We study the Cauchy problem of the damped wave equation ∂ 2 t u − ∆u + ∂tu = 0 and give sharp L p -L q estimates of the solution for 1 ≤ q ≤ p < ∞ (p = 1) with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial, s ≥ 0, and β = (n − 1)| 1 2 − 1 r |, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power 1 + 2r n , while it is known that the critical power 1+ 2 n belongs to the blow-up region when r = 1. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan by an ODE argument.

The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation

“…The Strichartz estimates for low frequency part. We have the L p -L q type estimates from [2] and [11]. These estimates are similar to those of the heat equation.…”

“…Matsumura's estimate (1.2) shows that the solution of (1.1) behaves like the solution of the heat equation and the wave equation in some sense. More precisely, the low frequency part of the solution to the damped wave equation behaves like the solution of the heat equation and the high frequency part behaves like the solution of the wave equation but decays exponentially (see [11] for another L p -L q estimate). For the heat equation and the wave equation, by using the L p -L q type estimates, we obtain the space-time estimates, what we call the Strichartz estimate.…”

Semilinear wave models with friction and viscoelastic damping

“…Remark Thanks to Ikeda et al, for m ∈[ m 0 ,2), m 0 >1, one may expect that may be included in the statements of Theorem . So, in this case, the lower bound for admissible p can be described by This shows a nice influence of additional regularity L m of the data on the solvability behavior of the model .…”

“…Remark 2. Thanks to Ikeda et al, 3 for m ∈ [m 0 , 2), m 0 > 1, one may expect that p = 1 + 2m n may be included in the statements of Theorem 1. So, in this case, the lower bound for admissible p can be described by…”

Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case

“…There are also large amount of literature of mathematical results about global existence of solutions, asymptotic behaviors of solutions, and blow-up phenomena to (1.1) in the opposite case b(t) −1 / ∈ L 1 (0, ∞) to (1.2), whose case is called effective damping or non-effective damping (see [25,31,38,29,30,26,2,3,34,1,4,15,12,33,36,16,23,13] and the references therein). However, there has been no result about the global existence of solutions to (1.1) in the overdamping case (1.2), especially there has not been known whether local energy solutions to (1.1) can be extended globally or not in the overdamping case (1.2).…”