Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density

A. V. Martynenko ; A. F. Tedeev

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 2, pp. 245-255

Résumé

The following quasilinear parabolic equation with a source term and an inhomogeneous density is considered: $$ \rho(x)\frac{\partial u}{\partial t}=\operatorname{div}(u^{m-1}|Du|^{\lambda-1}Du)+u^p. $$ The conditions on the parameters of the problem are found under which the solution to the Cauchy problem blows up in a finite time. A sharp universal (i.e., independent of the initial function) estimate of the solution near the blowup time is obtained.

Bibliographie

[1] Kamin S., Rosenau P., “Nonlinear diffusion in a finite mass medium”, Commyns. Pure and Appl. Math., 35 (1982), 113–127 | DOI | MR | Zbl

[2] Kamin S., Rosenau P., “Propagation of thermal waves in an inhomogeneous medium”, Communs. Pure and Appl. Math., 34 (1981), 831–852 | DOI | MR | Zbl

[3] Kamin S., Kersner R., “Disappearence of interfaces in finite time”, Meccanica, 28 (1993), 117–120 | DOI | Zbl

[4] Guedda M., Hilhorst D., Peletier M. A., “Disappearing interfaces in nonlinear diffusion”, Advances. Math. Sci. Appl., 7 (1997), 695–710 | MR | Zbl

[5] Galaktionov V. A., King J. R., “On the behaviour of blow-up interfaces for an inhomogeneous filtration equation”, J. Appl. Math., 57 (1996), 53–77 | MR | Zbl

[6] Kersner R., Reyes G., Tesei A., “On a class of nonlinear parabolic equations with variable density and absorption”, Advances Different. Equations, 7:2 (2002), 155–176 | MR | Zbl

[7] Tedeev A. F., “Usloviya suschestvovaniya i nesuschestvovaniya v tselom po vremeni kompaktnogo nositelya reshenii zadachi Koshi dlya kvazilineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii”, Sibirskii matem. zhurnal, 45:1 (2004), 189–200 | MR | Zbl

[8] Kurdyumov S. P., Kurkina E. S., “Spektr sobstvennykh funktsii dlya nelineinogo uravneniya teploprovodnosti s istochnikom”, Zh. vychisl. matem. i matem. fiz., 44:9 (2004), 1619–1637 | MR | Zbl

[9] Fujita H., “On the blowing up of solutions to the Cauchy problem for $u_t=\delta u+u^{1+\alpha}$”, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109–124 | MR | Zbl

[10] Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Samarskii A. A., “O neogranichennykh resheniyakh zadachi Koshi dlya parabolicheskogo uravneniya $u_t=\nabla(u^\sigma\nabla u)+u^\beta$”, Dokl. AN SSSR, 252:6 (1980), 1362–1364 | MR | Zbl

[11] Andreucci D., Tedeev A. F., “A Fujita type result for degenerate Neumann problem in domains with noncompact boundary”, J. Math. Analys and Appl., 231 (1999), 543–567 | DOI | MR | Zbl

[12] Andreucci D., Tedeev A. F., “Optimal bounds and blow-up phenomena for parabolic problems in narrowing domains”, Proc. Roy. Soc. Edinburg Sect. A, 128:6 (1998), 1163–1180 | MR | Zbl

[13] Deng K., Levine H. A., “The role of critical exponents in blow up theorems: The sequel”, J. Math. Analys and Appl., 243 (2000), 85–126 | DOI | MR | Zbl

[14] Samarskii A. A., Galaktionov B. A., Kurdyumov S. P., Mikhailov A. P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987 | MR

[15] Andreucci D., Tedeev A. F., “Universal bounds at the blow-up time for nonlinear parabolic equations”, Advances Different. Equat., 10:1 (2005), 89–120 | MR | Zbl

[16] Bernis F., “Existence results for doubly nonlinear higher order parabolic equations on unbounded domains”, Math. Ann., 279 (1988), 373–394 | DOI | MR | Zbl

[17] Alt H. W., Luckhaus S., “Quasilinear elliptic-parabolic differential equations”, Math. Z, 183 (1983), 311–341 | DOI | MR | Zbl

[18] Caffarelli L., Kohn R., Nirenberg L., “First order interpolation inequalities with weights”, Compositio Math., 53 (1984), 259–275 | MR | Zbl

[19] Ladyzhenskaya O. A., Solonnikov B. A., Uraltseva H. H., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967