Weighted $L^2-L^2$ estimate for wave equation and its applications

Abstract

In this work we establish a weighted $L^2-L^2$ estimate for inhomogeneous wave equation in 3-D, by introducing a Morawetz multiplier with weight of power $s(1<s<2)$, and then integrating on the light cones and $t$ slice. With this weighted $L^2-L^2$ estimate in hand, we may give a new proof of global existence for small data Cauchy problem of semilinear wave equation with supercritical power in 3-D. What is more, by combining the Huygens' principle for wave equations in 3-D, the global existence for semilinear wave equation with scale invariant damping in 3-D is established.

References in corpus (3)

• Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent
• A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent
• The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

Cited by in corpus (5)

• Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma
• A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms
• A competition on blow-up for semilinear wave equations with scale-invariant damping and nonlinear memory term
• Strauss exponent for semilinear wave equations with scattering space dependent damping
• Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution