Blow-up behaviour of one-dimensional semilinear parabolic equations. (English) Zbl 0813.35007

Summary: Consider the Cauchy problem \[ u_ t - u_{xx} - F(u) = 0; \quad x \in \mathbb{R}, \quad t > 0, \qquad u(x,0) = u_ 0(x); \quad x \in \mathbb{R} \] where \(u_ 0(x)\) is continuous, nonnegative and bounded, and \(F(u) = u^ p\) with \(p>1\), or \(F(u) = e^ u\). Assume that \(u\) blows up at \(x = 0\) and \(t = T > 0\). In this paper we describe the various possible asymptotic behaviours of \(u(x,t)\) as \((x,t) \to (0,T)\). Moreover, we show that if \(u_ 0(x)\) has a single maximum at \(x=0\) and is symmetric, \(u_ 0(x) = u_ 0 (-x)\) for \(x>0\), there holds
1) If \(F(u) = u^ p\) with \(p>1\), then \[ \begin{split} \lim_{t \uparrow T} u \biggl( \xi \bigl( (T-t) | \log (T - t) | \bigr)^{1/2}, t \biggr) \\ \times (T-t)^{1/(p-1)} = (p-1)^{-(1/(p-1))} \left[ 1 + {(p-1) \xi^ 2 \over 4p} \right]^{-(1/(p-1))} \end{split} \] uniformly on compact sets \(| \xi | \leq R\) with \(R>0\),
2) If \(F(u) = e^ u\), then \[ \lim_{t \uparrow T} \Bigl( u \biggl( \xi \bigl( (T-t) | \log (T-t) | \bigr)^{1/2}, t \biggr) + \log (T - t) \Bigr) = - \log \left[ 1 + {\xi^ 2 \over 4} \right] \] uniformly on compact sets \(| \xi | \leq R\) with \(R>0\).


35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
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[1] Angenent, S., The Zero Set of a Solution of a Parabolic Equation, J. reine angew Math., Vol. 390, 79-96 (1988) · Zbl 0644.35050
[2] Angenent, S. B.; Fiedler, B., The Dynamics of Rotating Waves in Scalar Reaction-Diffusion Equations, Trans. Amer. Math. Soc., Vol. 307, 545-568 (1988) · Zbl 0696.35086
[3] Aronson, D. G.; Weinberger, H. F., Multidimensional Nonlinear Diffusion arising in Population Genetics, Advances in Math., Vol. 30, 33-76 (1978) · Zbl 0407.92014
[4] Bebernes, J.; Bressan, A.; Eberly, D., A Description of Blow-up for the Solid Fuel Ignition Model, Indiana Univ. Math. J., Vol. 36, 131-136 (1987)
[5] Bressan, A., On the Asymptotic Shape of Blow-up, Indiana Univ. Math. J., Vol. 39, 947-960 (1990) · Zbl 0705.35014
[6] Chen, X. Y.; Matano, H.; Veron, L., Anisotropic Singularities of Solutions of Nonlinear Elliptic Equations in \(ℝ^2\), J. Fund. Anal., Vol. 83, 50-93 (1989)
[7] Cohen, P. J.; Lees, M., Asymptotic decay of Differential Inequalities, Pacific J. Math., Vol. 11, 1235-1249 (1961) · Zbl 0171.35002
[8] Dold, J., Analysis of the Early Stage of Thermal Runaway, Quart. J. Mech. Appl. Math., Vol. 38, 361-387 (1985) · Zbl 0569.76079
[9] Friedman, A.; McLeod, J. B., Blow-up of positive Solutions of Semilinear Heat Equations, Indiana Univ. Math. J., Vol. 34, 425-447 (1985) · Zbl 0576.35068
[10] Fujita, H., On the Blowing-up of Solutions of the Cauchy Problem for \(u_t = Δ u + u^{1+α}\), J. Fac. Sci. Univ. of Tokio, Section I, Vol. 13, 109-124 (1966) · Zbl 0163.34002
[11] Galaktionov, V. A.; Herrero, M. A.; Velázquez, J. J.L., The Space Structure near a Blow-up Point for Semilinear Heat Equations: a formal Approach, Soviet J. Comput. Math, and Math. Physics, Vol. 31, 399-411 (1991)
[12] Galakationov, V. A.; Posashkov, S. A., Application of new Comparison Theorems in the Investigation of Unbounded Solutions of nonlinear Parabolic Equations, Diff Urav., Vol. 22, 7, 1165-1173 (1986)
[13] Giga, Y.; Kohn, R. V., Asymptotically Self-Similar Blow-up of Semilinear Heat Equations, Comm. Pure Appl. Math., Vol. 38, 297-319 (1985) · Zbl 0585.35051
[14] Giga, Y.; Kohn, R. V., Characterizing Blow-up using Similarity Variables, Indiana Univ. Math., J., Vol. 36, 1-40 (1987) · Zbl 0601.35052
[15] Giga, Y.; Kohn, R. V., Nondegeneracy of Blow-up for Semilinear Heat Equations, Comm. Pure Appl. Math., Vol. 42, 845-884 (1989) · Zbl 0703.35020
[17] Herrero, M. A.; Velázquez, J. J.L., Flat Blow-up in One-Dimensional Semilinear Heat Equations, Differential and Integral Equations, Vol. 5, 973-997 (1992) · Zbl 0767.35036
[18] Lacey, A. A., The Form of Blow-up for Nonlinear Parabolic Equations, Proc. Royal Soc. Edinburgh, Vol. 98 A, 183-202 (1984) · Zbl 0556.35077
[19] Lax, P. D., A Stability Theorem for Solutions of Abstract Differential Equations, and its Application to the Study of the Local behaviour of Solutions of Elliptic Equations, Comm. Pure Appl. Math., Vol. 9, 747-766 (1956) · Zbl 0072.33004
[20] Liu, W., The Blow-up Rate of Solutions of Semilinear Heat Equations, J. Diff Equations, Vol. 77, 104-122 (1989) · Zbl 0672.35035
[21] Müller, C. E.; Weissler, F. B., Single Point Blow-up for a General Semilinear Heat Equation, Indiana Univ. Math., Vol. 34, 881-913 (1983) · Zbl 0597.35057
[22] Weissler, F. B., Single Point Blow-up of Semilinear Initial Value Problems, J. Diff. Equations, Vol. 55, 204-224 (1984) · Zbl 0555.35061
[23] Watson, N. A., Parabolic Equations on an Infinite Strip, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 127 (1988), Marcel Dekker
[24] Widder, D. V., The Heat Equation (1975), Academic Press: Academic Press New York · Zbl 0322.35041
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