Quenching for a degenerate parabolic problem due to a concentrated nonlinear source
Authors:
C. Y. Chan and X. O. Jiang
Journal:
Quart. Appl. Math. 62 (2004), 553-568
MSC:
Primary 35K60; Secondary 35K57, 35K65
DOI:
https://doi.org/10.1090/qam/2086046
MathSciNet review:
MR2086046
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Abstract: Let $q$, $a$, $T$, and $b$ be any real numbers such that $q \ge 0$, $a > 0$, $T > 0$, and $0 < b < 1$. This article studies the following degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at $b$: \[ {x^q}{u_t} - {u_{xx}} = {a^2}\delta \left ( x - b \right )f\left ( u\left ( x, t \right ) \right ) in \left ( 0, 1 \right ) \times \left ( 0, T \right ],\] \[ u\left ( x, 0 \right ) = 0 on \left [ 0, 1 \right ], u\left ( 0, t \right ) = u\left ( 1, t \right ) = 0 for \; 0 < t \le T,\] where $\delta \left ( x \right )$ is the Dirac delta function, $f$ is a given function such that ${\lim _{u \to {c^ - }}}f\left ( u \right ) = \infty$ for some positive constant $c$, and $f\left ( u \right )$ and $f’\left ( u \right )$ are positive for $0 \le u < c$. It is shown that the problem has a unique continuous solution $u$ before $max\left \{ {u\left ( x, t \right ) : 0 \le x \le 1} \right \}$ reaches ${c^ - }$, $u$ is a strictly increasing function of $t$ for $0 < x < 1$, and if $max\left \{ {u\left ( x, t \right ):0 \le x \le 1} \right \}$ reaches ${c^ - }$, then $u$ attains the value $c$ only at the point $b$. The problem is shown to have a unique ${a^*}$ such that a unique global solution $u$ exists for $a \le {a^*}$, and $max\left \{ {u\left ( x, t \right ):0 \le x \le 1} \right \}$ reaches ${c^ - }$ in a finite time for $a > {a^*}$; this ${a^*}$ is the same as that for $q = 0$. A formula for computing ${a^*}$ is given, and no quenching in infinite time is deduced.
- C. Y. Chan and W. Y. Chan, Existence of classical solutions for degenerate semilinear parabolic problems, Appl. Math. Comput. 101 (1999), no. 2-3, 125–149. MR 1678117, DOI https://doi.org/10.1016/S0096-3003%2898%2910002-4
- C. Y. Chan and Hans G. Kaper, Quenching for semilinear singular parabolic problems, SIAM J. Math. Anal. 20 (1989), no. 3, 558–566. MR 990863, DOI https://doi.org/10.1137/0520039
- C. Y. Chan and P. C. Kong, Quenching for degenerate semilinear parabolic equations, Appl. Anal. 54 (1994), no. 1-2, 17–25. MR 1382204, DOI https://doi.org/10.1080/00036819408840265
- C. Y. Chan and P. C. Kong, Channel flow of a viscous fluid in the boundary layer, Quart. Appl. Math. 55 (1997), no. 1, 51–56. MR 1433751, DOI https://doi.org/10.1090/qam/1433751
- C. Y. Chan and H. T. Liu, Does quenching for degenerate parabolic equations occur at the boundaries?, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 8 (2001), no. 1, 121–128. Advances in quenching. MR 1820670
- C. Y. Chan and H. Y. Tian, Single-point blow-up for a degenerate parabolic problem due to a concentrated nonlinear source, Quart. Appl. Math. 61 (2003), no. 2, 363–385. MR 1976376, DOI https://doi.org/10.1090/qam/1976376
- Keng Deng and Catherine A. Roberts, Quenching for a diffusive equation with a concentrated singularity, Differential Integral Equations 10 (1997), no. 2, 369–379. MR 1424817
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Karl E. Gustafson, Introduction to partial differential equations and Hilbert space methods, 2nd ed., John Wiley & Sons, Inc., New York, 1987. MR 881383
- H. L. Royden, Real analysis, 3rd ed., Macmillan Publishing Company, New York, 1988. MR 1013117
- Karl R. Stromberg, Introduction to classical real analysis, Wadsworth International, Belmont, Calif., 1981. Wadsworth International Mathematics Series. MR 604364
- G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110
C. Y. Chan and W. Y. Chan, Existence of classical solutions for degenerate semilinear parabolic problems, Appl. Math. Comput. 101, 125-149 (1999)
C. Y. Chan and H. G. Kaper, Quenching for semilinear singular parabolic problems, SIAM J. Math Anal 20, 558-566 (1989)
C. Y. Chan and P. C. Kong, Quenching for degenerate semilinear parabolic equations, Appl. Anal. 54, 17-25 (1994)
C. Y. Chan and P. C. Kong, Channel flow of a viscous fluid in the boundary layer, Quart. Appl. Math 55, 51-56 (1997)
C. Y. Chan and H. T. Liu, Does quenching for degenerate parabolic equations occur at the boundaries?, Dynam. Contin. Discrete Impuls. Systems (Series A) 8, 121-128 (2001)
C. Y. Chan and H. Y. Tian, Single-point blow-up for a degenerate parabolic problem due to a concentrated nonlinear source, Quart. Appl. Math. 61, 363-385 (2003).
K. Deng and C. A. Roberts, Quenching for a diffusive equation with a concentrated singularity, Differential Integral Equations 10, 369-379 (1997)
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964, pp. 39 and 49
K. E. Gustafson, Introduction to Partial Differential Equations and Hilbert Space Methods, 2nd ed., John Wiley &, Sons, New York, NY, 1987, p. 176
H. L. Royden, Real Analysis, 3rd ed., Macmillan Publishing Co., New York, NY, 1988, p. 87
K. R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth International Group, Belmont, CA, 1981, pp. 328, 352, and 380
G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, New York, NY, 1958, p. 506
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© Copyright 2004
American Mathematical Society