Interior derivative blow-up for quasilinear parabolic equations.

*(English)*Zbl 0878.35015The standard theory of quasilinear second order parabolic equations [O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Am. Math. Soc. (1968; Zbl 0174.15403), G. M. Lieberman, Second Order Parabolic Differential Equations (World Scientific 1996)] shows that a key step in the existence program is the verification of certain a priori estimates for all possible solutions, and there are four basic estimates: on the maximum of the solution, on the maximum of its gradient on the boundary of the domain, on the maximum of the gradient inside the domain, and on the HĂ¶lder norm of the gradient. It has long been known that the boundary gradient estimate is crucial in the existence program. Examples due to Fillipov and Serrin (and modified to the parabolic case by the reviewer) show that when the hypotheses implying this estimate are violated, then the problem need not have a solution for all time. In all these examples, the solution has an infinite derivative on the boundary at some finite time.

This paper shows a new kind of nonexistence result: the solution fails to exist for all time because the derivative becomes infinite at some interior point in finite time. The examples here are all for a problem with one space dimension, and the methods are based on the author’s previous work on curve shortening. Since this paper’s publication, several other works have appeared on this same topic. We mention, in particular, the paper of S. B. Angenent and M. Fila [Differ. Integral Equ. 9 , 865-877 (1996. Zbl 0864.35052)], which is listed in the bibliography and a new paper by K. Asai and N. Ishimura [On the interior derivative blow-up for the curvature evolution of capillary surfaces, Proc. Am. Math. Soc. (to appear)]. The latter work is the only one to give a truly higher dimensional example of interior gradient blow-up.

This paper shows a new kind of nonexistence result: the solution fails to exist for all time because the derivative becomes infinite at some interior point in finite time. The examples here are all for a problem with one space dimension, and the methods are based on the author’s previous work on curve shortening. Since this paper’s publication, several other works have appeared on this same topic. We mention, in particular, the paper of S. B. Angenent and M. Fila [Differ. Integral Equ. 9 , 865-877 (1996. Zbl 0864.35052)], which is listed in the bibliography and a new paper by K. Asai and N. Ishimura [On the interior derivative blow-up for the curvature evolution of capillary surfaces, Proc. Am. Math. Soc. (to appear)]. The latter work is the only one to give a truly higher dimensional example of interior gradient blow-up.

Reviewer: G.M.Lieberman (Ames)

##### MSC:

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35K55 | Nonlinear parabolic equations |