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Blow-up behavior of one-dimensional semilinear parabolic equations

Herrero, Miguel A. and Velázquez, J.J. L. (1993) Blow-up behavior of one-dimensional semilinear parabolic equations. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 10 (2). pp. 131-189. ISSN 0294-1449

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Abstract

Consider the Cauchy problem u(t) - u(xx) - F(u) = 0; x is-an-element-of R, t>0 u(x, 0) = u0(x); x is-an-element-of R where u0 (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 shall describe the various possible asymptotic behaviours of u(x, t) as (x, t) --> (0, T). Moreover, we shall show that if u0(x) has a single maximum at x = 0 and is symmetric, u0(x) = u0(- x) for x > 0, there holds 1)If F(u) = u(p) with p > 1, then lim u(xi((T - t)\log (T - t)\)1/2, t) t up T x(T - t)1/(p - 1) = (p - 1) - (1/(p - 1)) [1 + (p - 1)xi2/4p] - 1/(p - 1)) uniformly on compact sets \xi\ less-than-or-equal-to R with R > 0, 2) If F(u) = e(u), then lim (u(xi((T - t)\log (T - t)\)1/2, t) + log(T - t)) = - log [1 + xi2/4] t up T uniformly on compact sets \xi\ less-than-or-equal-to R with R>0.

Item Type:Article
Uncontrolled Keywords:Semilinear diffusion equations; blow-up; asymptotic behavior of solutions; heat-equations
Subjects:Sciences > Mathematics > Differential equations
ID Code:17095
References:

S. Angenent, The Zero Set of a Solution of a Parabolic Equation, J. reine angew Math., Vol. 390, 1988, pp. 79-96.

S.B. Angenent and B. Fiedler, The Dynamics of Rotating Waves in Scalar Reaction-Diffusion Equations, Trans. Amer. Math. Soc., Vol. 307, 1988, pp. 545-568.

D.G. Aronson and H.F. Weinberger, Multidimensional Nonlinear Diffusion arising in Population Genetics, Advances in Math., Vol. 30, 1978, pp. 33-76.

J. Bebernes, A. Bressan and D. Eberly, A Description of Blow-up for the Solid Fuel Ignition Model, Indiana Univ. Math. J., Vol. 36, 1987, pp. 131-136.

A. Bressan, On the Asymptotic Shape of Blow-up, Indiana Univ. Math. J., Vol. 39, 1990, pp. 947-960.

X.Y. Chen, H. Matano and L. Veron, Anisotropic Singularities of Solutions of Nonlinear Elliptic Equations in R2, J. Funct. Anal., Vol. 83, 1989, pp. 50-93.

P.J. Cohen and M. Lees, Asymptotic decay of Differential Inequalities, Pacific J. Math, Vol. 11, 1961, pp. 1235-1249.

J. Dold, Analysis of the Early Stage of Thermal Runaway, Quart. J. Mech. Appl. Math., Vol. 38, 1985, pp. 361-387.

A. Friedman and J.B. McLeod, Blow-up of positive Solutions of Semilinear Heat Equations, Indiana Univ. Math. J., Vol. 34, 1985, pp. 425-447.

H. Fujita, On the Blowing-up of Solutions of the Cauchy Problem for ut = Δu + u1+α, J. Fac. Sci. Univ. of Tokio, Section I, Vol. 13, 1966, pp. 109-124.

V.A. Galaktionov, M.A. Herrero and J.J.L. Velázquez, The Space Structure near a Blow-up Point for Semilinear Heat Equations: a formal Approach, Soviet J. Comput. Math. and Math. Physics, Vol. 31, 1991, pp. 399-411.

V.A. Galakationov and S.A. Posashkov, Application of new Comparison Theorems in the Investigation of Unbounded Solutions of nonlinear Parabolic Equations, Diff. Urav., Vol. 22, 7, 1986, pp. 1165-1173.

Y. Giga and R.V. Kohn, Asymptotically Self-Similar Blow-up of Semilinear Heat Equations, Comm. Pure Appl. Math., Vol. 38, 1985, pp. 297-319.

Y. Giga and R.V. Kohn, Characterizing Blow-up using Similarity Variables, Indiana Univ. Math., J., Vol. 36, 1987, pp. 1-40.

Y. Giga and R.V. Kohn, Nondegeneracy of Blow-up for Semilinear Heat Equations, Comm. Pure Appl. Math., Vol. 42, 1989, pp. 845-884.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes in Mathematics No. 840, 1981.

M.A. Herrero and J.J.L. Velázquez, Flat Blow-up in One-Dimensional Semilinear Heat Equations, Differential and Integral Equations, Vol. 5, 1992, pp. 973-997.

A. A. LACEY, The Form of Blow-up for Nonlinear Parabolic Equations, Proc. Royal Soc. Edinburgh, Vol. 98 A, 1984, pp. 183-202.

P.D. Lax, 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, 1956, pp. 747-766.

W. Liu, The Blow-up Rate of Solutions of Semilinear Heat Equations, J. Diff. Equations Vol. 77, 1989, pp. 104-122.

C.E. Müller and F.B. Weissler, Single Point Blow-up for a General Semilinear Heat Equation, Indiana Univ. Math., J., Vol. 34, 1983, pp. 881-913.

F.B. Weissler, Single Point Blow-up of Semilinear Initial Value Problems, J. Diff. Equations, Vol. 55, 1984, pp. 204-224.

N.A. Watson, Parabolic Equations on an Infinite Strip, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 127, Marcel Dekker, 1988.

D.V. Widder, The Heat Equation, Academic Press, New York, 1975.

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