$L^p$-$L^q$ 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 \begin{align*} \partial_{t}^2 u - Δu + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q\le p<\infty (p\ne 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\cap H_r^{\beta })\times (H^{s-1}\cap L^r)$ with $r\in (1,2]$, $s\ge 0$, and $\beta =(n-1)|\frac{1}{2}-\frac{1}{r}|$, 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 $1+\frac{2r}{n}$, while it is known that the critical power $1+\frac{2}{n}$ 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 and blow-up rates by an ODE argument.

44 pages, the statement and the proof of Theorem 1.3 are corrected, typos are corrected

Cited by in corpus (7)

• On the critical exponent and sharp lifespan estimates for semilinear damped wave equations with data from Sobolev spaces of negative order
• Sharp lifespan estimates for semilinear fractional evolution equations with critical nonlinearity
• Decay property of solutions to the wave equation with space-dependent damping, absorbing nonlinearity, and polynomially decaying data
• The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation
• Large time behavior of solutions to the Cauchy problem for the BBM-Burgers equation
• Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data
• Existence of solutions to the semilinear damped wave equation with non-$L^2$ slowly decaying data : polynomial nonlinearity case