The rescaling method for some quasilinear wave equations with the divergence forms of the nonlinearity

Abstract

In this paper, we will study the general kind of quasilinear wave equation □u=u2+∑i,j=132bij(uuxi)xj+∑i=132ciuuxi,inR3×[0,+∞). It is a special term of the divergence forms, but this is the first attempt to this direction without any non-local term which comes from the derivative loss. We obtain that for the compactly supported smooth initial values, the solution must blow up in finite time if the initial data are nonnegative and positive somewhere no matter how small the initial data are. The originality of the paper is the choice of “rescaled” test function (2.3)(One can refer to Section 2 for details). And also we give the sharp lifespan estimate of solution for the problem. This solves a part of the famous Strauss conjecture with regard to the general kind of quasilinear wave equations with subcritical exponent p=2 in three space dimensions.