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

ABSTRACT

The motivation of this paper is to extend the recent result of the author and his tutor for the solution of the critical semilinear wave equations of the form $\square u=u^2$ in four space dimensions to the general kind of quasilinear wave equations $\square u=u^2+{\sum }_{i=1}^4{a}_{i}u{u}_{x_i}+{\sum }_{i,j=1}^4{a}_{\mathit{ij}}{(u{u}_{x_i})}_{x_j}$ in $R^4\times [0,+\infty )$. We will prove 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, and also we give the sharp lifespan estimate of solutions for the problem. This solves a part of the famous Strauss’ conjecture with regard to the general kind of quasilinear wave equations in the case of critical exponent and in four space dimensions. The originality of the paper is the choice of ‘rescaled’ test function (2.8) (One can refer to Section 2 for details). The divergence form of the nonlinearity provides spatial derivatives to the test functions. 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 due to high dimensions. Thanks to the rescaling, spatial derivatives yield smallness of the nonlinearity except for main part $u^2$ among integration by parts in the functional method. It makes also positiveness of the nonlinearity. However, this combination of rescaled test functions and the divergence form of nonlinearities is a new idea.

KEYWORDS:

• Quasilinear wave equation
• critical exponent
• Cauchy problem
• blow up
• lifespan

AMS SUBJECT CLASSIFICATIONS:

• 35L45
• 35L60

Acknowledgements

The authors would like to thank their tutor Professor Zhou Yi for his guidance and encouragement, and thank Professor Li Ta-tsien and Professor Lei Zhen for their helpful suggestions and comments.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant number 11301489], the Distinguished Youth Science Foundation of Shanxi Province [grant number 2015021001], the Outstanding Youth Foundation of North University of China [grant number JQ201604], the Youth Academic Leaders Support Program of North University of China.