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 powerand 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.

Critical exponent for nonlinear damped wave equations with non-negative potential in 3D

“…The assumption about the radial symmetry posed in [1] was removed by Ikeda, Sobajima [8] for the blow-up part (actually, they treated more general damping term µ(1 + t) −1 ∂ t u with µ > 0), and by Kato, Sakuraba [10] and Lai [12] for the existence part, independently. Now we turn back to the case where the coefficient of the damping term is a function of spatial variabletime variavles.…”

A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale‐invariant lower order terms

“…= (µ − 1) 2 − 4ν 2 has a strong influence on some properties of solutions to (2) and to the corresponding homogeneous linear equation. According to [3,40,5,4,39,22,30,27,21,13,31,37,38,28,29,6,35,16,20] for δ 0 the model in (2) is somehow an intermediate model between the semilinear free wave equation and the semilinear classical damped equation, whose critical exponent is p Fuj (n + µ−1 2 − √ δ…”

Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part

“…The value of δ has a strong influence on some properties of solutions to (1.7) and to the corresponding homogeneous linear equation. According to [4,57,6,5,56,37,46,43,27,17,47,54,55,44,45,7,48,21,26] for δ 0 the model in (1.7) is somehow an intermediate model between the semilinear free wave equation and the semilinear classical damped equation, whose critical exponent is p Fuj (n + α − 1) for δ ≥ (n + 1) 2 , where α is defined analogously as in (1.3), and seems reasonably to be p 0 (n + µ) for small values of delta. In this paper we will deal with the system (1.1) and we will investigate how the interaction between the powers p, q in the nonlinearities provides either the global in time existence of the solution or the blow-up in finite time.…”