Blow-up of solutions of nonlinear wave equations in three space dimensions

Abstract

Solutions u(x, t) of the inequality 0 u > A I u I for x e R3, t > 0 are considered, where 3 is the d'Alembertian, and A,p are constants with A > 0,1 < p < 1 + V . It is shown that the support of u is compact and contained in the cone 0 < t < to -Ix -xO-, if the "initial data" u(x, 0), ut(x, 0) have their support in the ball I x -x < to. In particular, "global" solutions of 3 u = A ju I P with initial data of compact support vanish identically. On the other hand, for A > 0, p > 1 + X"_2, global solutions of 3 u = A ju I P exist, if the initial data are of compact support and "sufficiently" small. This paper is concerned with global existence of solutions u(x, t) = u(x1, x2, X3, t) of a nonlinear wave equation of the form Ou = 0(u) [1] or of an inequality of the form [2], of class C2 in the half-space x e R3, t > 0, for which f E C3, g E C2 for x e R.3 "Blow-up" consists in nonexistence of a global solution for given fg,k. In that case, instead of global solutions there may still exist local solutions u defined for x E R3 and sufficiently small t. We denote by uO(x, t) the solution of the linear wave equation 0u0= 0 [4]

Small solutions to semi-linear wave equations with radial data of critical regularity

“…We remark that the condition p > p 0 (n) (n = 2, 3) in Theorem 1.1 is just the same as in the classics of John [9] for n = 3 and Glassey [4] for n = 2 concerning the existence of global C 2 -solutions with small, smooth data of compact support. It was also proved by John (n = 3) and Glassey (n = 2) that there exist finite-time blow-up solutions even for small, smooth data of compact support if λ > 0 and p < p 0 (n) [9], [3]. (See, e.g., Schaeffer [15] for the blow up for p = p 0 (n), n = 2, 3.)…”

Exponential Growth for a Fractionally Damped Wave Equation

“…Roughly speaking, if the initial data u 0 and u 1 belong to an "unstable set", then the associated solution blows up in a finite time (see [4,14,19]). Then, solutions are shown to blow up also for small initial data provided that the exponent p lies in some "critical range" (see [5,6,13,34]). It is often proved that solutions are bounded and so exist globally beyond a critical power.…”

On the new critical exponent for the nonlinear Schrödinger equations

“…To reveal it, a special oscillating structure of the nonlinearity should be taken into account. We note that exponent p s (n) was discovered in paper [6], concerning the nonlinear wave equation with nonlinearity |u| p , where the global existence of small solutions was proved for the case of p > p s (n − 1) and the blow up of solutions was shown for p < p s (n − 1) , when n = 3. This result was generalized to the space dimensions n ≥ 2 in papers [3][4][5]13].…”