Blowup in diffusion equations: a survey

This paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems.

[1]  A. Stuart,et al.  On the computation of blow-up , 1990, European Journal of Applied Mathematics.

[2]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[3]  Catherine Sulem,et al.  Numerical simulation of singular solutions to the two‐dimensional cubic schrödinger equation , 1984 .

[4]  Jerrold Bebernes,et al.  Mathematical Problems from Combustion Theory , 1989 .

[5]  A. Lacey.  The form of blow-up for nonlinear parabolic equations , 1984, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Victor A. Galaktionov,et al.  Continuation of blowup solutions of nonlinear heat equations in several space dimensions , 1997 .

[7]  Jong-Shenq Guo,et al.  The blow-up behavior of the solution of an integrodifferential equation , 1992, Differential and Integral Equations.

[8]  Michael Grinfeld,et al.  Deciphering singularities by discrete methods , 1994 .

[9]  Weizhang Huang,et al.  Moving mesh partial differential equations (MMPDES) based on the equidistribution principle , 1994 .

[10]  W. Olmstead,et al.  Growth rates for blow-up solutions of nonlinear Volterra equations , 1996 .

[11]  D. Sattinger.  Topics in stability and bifurcation theory , 1973 .

[12]  R. Kohn,et al.  A rescaling algorithm for the numerical calculation of blowing-up solutions , 1988 .

[13]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[14]  Howard A. Levine,et al.  Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=−Au+ℱ(u) , 1973 .

[15]  V. A. Galaktionov,et al.  The conditions for there to be no global solutions of a class of quasilinear parabolic equations , 1982 .

[16]  T. Geveci,et al.  Numerical experiments with a nonlinear evolution equation which exhibits blow-up , 1992 .

[17]  F. Weissler,et al.  Self-similar subsolutions and blow-up for nonlinear parabolic equations , 1997 .

[18]  B. Gidas,et al.  Symmetry and related properties via the maximum principle , 1979 .

[19]  Catherine Bandle,et al.  On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains , 1989 .

[20]  A. Friedman.  Blow-up of solutions of nonlinear parabolic equations , 1988 .

[21]  H. Hattori,et al.  Global existence and blowup for a semilinear integral equation , 1990 .

[22]  Weizhang Huang,et al.  A Simple Adaptive Grid Method in Two Dimensions , 1994, SIAM J. Sci. Comput..

[23]  J. W. Dold,et al.  Blow-up in a System of Partial Differential Equations with Conserved First Integral. Part II: Problems with Convection , 1994, SIAM J. Appl. Math..

[24]  A. S. Kalashnikov.  Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations , 1987 .

[25]  A. Friedman,et al.  Blow-up of positive solutions of semilinear heat equations , 1985 .

[26]  Masahisa Tabata.  A finite difference approach to the number of peaks of solutions for semilinear parabolic problems , 1980 .

[27]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems IV: nonlinear problems , 1995 .

[28]  Y. Qi.  The critical exponents of parabolic equations and blow-up in Rn , 1998, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[29]  Haim Brezis,et al.  Blow up for $u_t-\Delta u=g(u)$ revisited , 1996, Advances in Differential Equations.

[30]  Lawrence E. Payne,et al.  DECAY BOUNDS FOR SOLUTIONS OF SECOND ORDER PARABOLIC PROBLEMS AND THEIR DERIVATIVES , 1995 .

[31]  Marie-Noëlle Le Roux,et al.  Semidiscretization in time of nonlinear parabolic equations with blowup of the solution , 1994 .

[32]  Michel Chipot,et al.  Some blowup results for a nonlinear parabolic equation with a gradient term , 1989 .

[33]  P Baras,et al.  Complete blow-up after Tmax for the solution of a semilinear heat equation , 1987 .

[34]  Ryuichi Suzuki,et al.  Critical exponent and critical blow-up for quasilinear parabolic equations , 1997 .

[35]  H. Bellout.  Blow-up of solutions of parabolic equations with nonlinear memory , 1987 .

[36]  Howard A. Levine,et al.  The Role of Critical Exponents in Blowup Theorems , 1990, SIAM Rev..

[37]  Isoperimetric inequalities for a class of nonlinear parabolic equations , 1976 .

[38]  On blow-up solutions of semilinear parabolic equations ; Analytical and numerical studies = 半線型放物型方程式の爆発解について ; 解析的ならびに数値的研究 , 1988 .

[39]  M N Le Roux.  Semi-discretization in time of a fast diffusion equation , 1989 .

[40]  Peter R. Turner,et al.  Topics in Numerical Analysis , 1982 .

[41]  P. Lions,et al.  Solutions globales d'équations de la chaleur semi linéaires , 1984 .

[42]  Andrew M. Stuart,et al.  Blowup in a Partial Differential Equation with Conserved First Integral , 1993, SIAM J. Appl. Math..

[43]  Andrew Alfred Lacey,et al.  Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition , 1988 .

[44]  O. Karakashian,et al.  Computations of blow-up and decay for periodic solutions of the generalized Korteweg-de Vries-Burgers equation , 1992 .

[45]  Haim Brezis,et al.  A nonlinear heat equation with singular initial data , 1996 .

[46]  A. P. Mikhailov,et al.  Blow-Up in Quasilinear Parabolic Equations , 1995 .

[47]  H. Brunner,et al.  Numerical analysis of semilinear parabolic problems with blow-up solutions. , 1994 .

[48]  W. E. Olmstead,et al.  Volterra Equations which Model Explosion in a Diffusive Medium , 1993 .

[49]  P. Mottoni,et al.  Attractivity properties of nonnegative solutions for a class of nonlinear degenerate parabolic problems , 1984 .

[50]  Ryuichi Suzuki.  Critical Blow-Up for Quasilinear Parabolic Equations in Exterior Domains , 1996 .

[51]  Robert D. Russell,et al.  Moving Mesh Methods for Problems with Blow-Up , 1996, SIAM J. Sci. Comput..

[52]  Claes Johnson,et al.  Computational Differential Equations , 1996 .

[53]  M. Fila.  Boundedness of Global Solutions of Nonlinear Diffusion Equations , 1992 .

[54]  H. Weinberger,et al.  Maximum principles in differential equations , 1967 .

[55]  Yoshikazu Giga,et al.  Nondegeneracy of blowup for semilinear heat equations , 1989 .

[56]  F. Weissler.  Semilinear evolution equations in Banach spaces , 1979 .

[57]  V. Galaktionov,et al.  Difference solutions of a class of quasilinear parabolic equations, I , 1983 .

[58]  L. A. Peletier,et al.  Stabilization of solutions of a degenerate nonlinear diffusion problem , 1982 .

[59]  J. C. López-Marcos,et al.  Blow-up for semidiscretizations of reaction-diffusion equations , 1996 .

[60]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[61]  Weizhang Huang,et al.  A moving collocation method for solving time dependent partial differential equations , 1996 .

[62]  J. M. Sanz-Serna,et al.  The Numerical Study of Blowup with Application to a Nonlinear Schrödinger Equation , 1992 .

[63]  A. Stuart,et al.  A note on uniform in time error estimates for approximations to reaction-diffusion equations , 1992 .

[64]  A. Peirce,et al.  The blowup property of solutions to some diffusion equations with localized nonlinear reactions , 1992 .

[65]  K. Hayakawa,et al.  On nonexistence of global solutions of some semilinear parabolic equations , 1973 .

[66]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[67]  Weizhang Huang,et al.  Analysis Of Moving Mesh Partial Differential Equations With Spatial Smoothing , 1997 .

[68]  Tomoyasu Nakagawa,et al.  Blowing up of a finite difference solution tout = uxx + u2 , 1975 .

[69]  Bei Hu,et al.  Remarks on the blowup estimate for solution of the heat equation with a nonlinear boundary condition , 1996, Differential and Integral Equations.

[70]  J. Serrin,et al.  Nonlinear Diffusion Equations and Their Equilibrium States II , 1988 .

[71]  C. Bandle,et al.  The positive radial solutions of a class of semilinear elliptic equations. , 1989 .

[72]  Wei-Ming Ni,et al.  On the asymptotic behavior of solutions of certain quasilinear parabolic equations , 1984 .

[73]  Peter Meier,et al.  On the critical exponent for reaction-diffusion equations , 1990 .