On the blow up of $u_ t$ at quenching

Authors:
Keng Deng and Howard A. Levine

Journal:
Proc. Amer. Math. Soc. **106** (1989), 1049-1056

MSC:
Primary 35B40; Secondary 35K55

DOI:
https://doi.org/10.1090/S0002-9939-1989-0969520-0

MathSciNet review:
969520

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Abstract: Let $\Omega$ be a bounded convex domain in ${{\mathbf {R}}^n}$ with smooth boundary. We consider the problems $\left ( C \right ):{u_t} = \Delta u + \varphi \left ( u \right )$ in $\Omega \times \left ( {0,T} \right )$, while $u = 0$ on $\partial \Omega \times \left ( {0,T} \right )$ and $u\left ( {x,0} \right ) = {u_0}\left ( x \right )$. Here $\varphi \left ( u \right ):\left ( { - \infty ,A} \right ) \to \left ( {0,\infty } \right )\left ( {A > 0} \right )$ satisfies $\varphi ’\left ( u \right ) \geq 0,\varphi ''\left ( u \right ) \geq 0$, and ${\lim _{u \to {A^ - }}}\varphi \left ( u \right ) = + \infty$, while ${u_0}$ satisfies $\Delta {u_0}\left ( x \right ) + \varphi \left ( {{u_0}\left ( x \right )} \right ) \geq 0$. We show that if $u$ quenches (reaches $A$ in finite time), then the quenching points are in a compact subset of $\Omega$ and ${u_t}$ blows up. We also extend the result to the third boundary value problem.

- Hideo Kawarada,
*On solutions of initial-boundary problem for $u_{t}=u_{xx}+1/(1-u)$*, Publ. Res. Inst. Math. Sci.**10**(1974/75), no. 3, 729–736. MR**0385328**, DOI https://doi.org/10.2977/prims/1195191889 - Andrew Acker and Wolfgang Walter,
*The quenching problem for nonlinear parabolic differential equations*, Ordinary and partial differential equations (Proc. Fourth Conf., Univ. Dundee, Dundee, 1976) Springer, Berlin, 1976, pp. 1–12. Lecture Notes in Math., Vol. 564. MR**0604032** - Andrew Acker and Wolfgang Walter,
*On the global existence of solutions of parabolic differential equations with a singular nonlinear term*, Nonlinear Anal.**2**(1978), no. 4, 499–504. MR**512487**, DOI https://doi.org/10.1016/0362-546X%2878%2990057-3 - Howard A. Levine and John T. Montgomery,
*The quenching of solutions of some nonlinear parabolic equations*, SIAM J. Math. Anal.**11**(1980), no. 5, 842–847. MR**586912**, DOI https://doi.org/10.1137/0511075 - Howard A. Levine,
*The quenching of solutions of linear parabolic and hyperbolic equations with nonlinear boundary conditions*, SIAM J. Math. Anal.**14**(1983), no. 6, 1139–1153. MR**718814**, DOI https://doi.org/10.1137/0514088 - Howard A. Levine,
*The phenomenon of quenching: a survey*, Trends in the theory and practice of nonlinear analysis (Arlington, Tex., 1984) North-Holland Math. Stud., vol. 110, North-Holland, Amsterdam, 1985, pp. 275–286. MR**817500**, DOI https://doi.org/10.1016/S0304-0208%2808%2972720-8
---, - Andrew F. Acker and Bernhard Kawohl,
*Remarks on quenching*, Nonlinear Anal.**13**(1989), no. 1, 53–61. MR**973368**, DOI https://doi.org/10.1016/0362-546X%2889%2990034-5 - Avner Friedman and Bryce McLeod,
*Blow-up of positive solutions of semilinear heat equations*, Indiana Univ. Math. J.**34**(1985), no. 2, 425–447. MR**783924**, DOI https://doi.org/10.1512/iumj.1985.34.34025 - B. Gidas, Wei Ming Ni, and L. Nirenberg,
*Symmetry and related properties via the maximum principle*, Comm. Math. Phys.**68**(1979), no. 3, 209–243. MR**544879**
M. Chipot and F. B. Weissler, - C. Y. Chan and Man Kam Kwong,
*Quenching phenomena for singular nonlinear parabolic equations*, Nonlinear Anal.**12**(1988), no. 12, 1377–1383. MR**972406**, DOI https://doi.org/10.1016/0362-546X%2888%2990085-5
J. Guo,

*Quenching, nonquenching, and beyond quenching for solutions of some parabolic equations*, Ann. Mat. Pura Appl. (in press).

*Some blow up results for a nonlinear parabolic equation with a gradient term*, IMA Preprint Series no. 298.

*On the quenching behavior of the solution of a semilinear parabolic equation*, J. Math. Anal. Appl. (in print). ---,

*On the semilinear elliptic equation*, $\Delta w - \frac {1}{2}y \cdot \nabla w + \lambda w - {w^{ - \beta }} = 0$ in ${R^N}$, (in print).

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© Copyright 1989
American Mathematical Society