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Blow-up at the boundary for degenerate semilinear parabolic equations. (English) Zbl 0733.35009
Consider the semilinear problem: $xu\sb t=u\sb{xx}+u\sp p$ in $\Omega\times (0,\infty)$, $u(0,t)=u(1,t)=0$ for $t>0$, $u(x,0)=u\sb 0(x)\ge 0$ for $x\in \Omega$ where $\Omega$ is the unit interval (0,1), $p>1$ and the initial function $u\sb 0\in C\sp 1({\bar \Omega})$ with $u\sb 0(0)=u\sb 0(1)=0$. The author proves the theorems: (i) Suppose that $1<p\le 2$ and the solution u blows up at $t=T$. If $(\partial /\partial x)(u\sb 0(x)/x)\le 0$ for $x\in (0,1)$, then $S=\{0\}$, where S is the set of blow-up points. (ii) Suppose that $1<p\le 2$, $(\partial /\partial x)(u\sb 0(x)/x)\le 0$ and u blows up at $t=T.$ Then given $\delta >0$, there exists a $C\sb 2>0$ such that $s\sb 2(t)\le C\sb 2(T-t)\sp{1/3+\delta}$. Under the additional assumption $u\sb 0''+u\sb 0\sp p\ge 0$, there exists a $C\sb 1>0$ such that $s\sb 1(t)\ge C\sb 1(T-t)\sp{1/3}$ for all $t\in (0,T)$, where s(t) be any point such that $u(s(t),t)=\sup\sb{x\in \Omega}u(x,t)$, and $s\sb 1(t)$ is the infimum and $s\sb 2(t)$ the supremum over all such s.

##### MSC:
 35B40 Asymptotic behavior of solutions of PDE 35K65 Parabolic equations of degenerate type 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
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##### References:
 [1] Abramowitz, M., & A. Stegun (1965) ?Handbook of mathematical functions?, Dover Publications, New York. [2] Floater, M. S., (1988) ?Blow-up of solutions to nonlinear parabolic equations and systems?, D. Phil. thesis, Univ. of Oxford. [3] Friedman, A. (1964) ?Partial differential equations of parabolic type?, Prentice-Hall, Englewood Cliffs, New Jersey. · Zbl 0144.34903 [4] Friedman, A., & J. B. McLeod (1985) ?Blow-up of positive solutions of semilinear heat equations?, Indiana Univ. Math. J. 34, 425-447. · Zbl 0576.35068 · doi:10.1512/iumj.1985.34.34025 [5] Kaplan, S. (1963) ?On the growth of solutions of quasilinear parabolic equations?, Comm. Pure Appl. Math. 16, 305-330. · Zbl 0156.33503 · doi:10.1002/cpa.3160160307 [6] Lacey, A. A. (1984) ?The form of blow-up for nonlinear parabolic equations?, Proc. Roy. Soc. Edin. 98, 183-202. · Zbl 0556.35077 [7] Ladyzenskaya, O. A., V. A., Solonnikov & N. N. Ural’ceva (1968) ?Linear and quasilinear equations of parabolic type?, Amer. Math. Soc. Translations of Mathematical Monographs, Providence. [8] Mueller, C. E., & F. B. Weissler (1985) ?Single point blow-up for a general semilinear heat equation?, Indiana Univ. Math. J. 34, 881-913. · Zbl 0597.35057 · doi:10.1512/iumj.1985.34.34049 [9] Ockendon, H. (1979) ?Channel flow with temperature-dependent viscosity and internal viscous dissipation?, J. Fluid Mech. 93, 737-746. · Zbl 0406.76043 · doi:10.1017/S0022112079002007 [10] Sattinger, D. H. (1972) ?Monotone methods in nonlinear elliptic and parabolic boundary value problems?, Indiana Univ. Math. J. 21, 979-1000. · Zbl 0223.35038 · doi:10.1512/iumj.1972.21.21079 [11] Stuart, A., & M. S. Floater (1990) ?On the computation of blow-up?, Euro. J. Appl. Math. 1, 47-71. · Zbl 0701.76038 · doi:10.1017/S095679250000005X