The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation

— We review the main mathematical questions posed in blow-up problems for reactiondiffusion equations and discuss results of the author and collaborators on the subjects of continuation of solutions after blow-up, existence of transient blow-up solutions (so-called peaking solutions) and avalanche formation as a mechanism of complete blow-up.

[1]  F. Quirós,et al.  COMPLETE BLOW-UP AND THERMAL AVALANCHE FOR HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS , 2002 .

[2]  Ryuichi Suzuki.  Blow-up problems for quasilinear parabolic equations , 1997 .

[3]  Minkyu Kwak,et al.  SELF-SIMILAR SOLUTIONS OF A SEMILINEAR HEAT EQUATION , 2004 .

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

[5]  A. Friedman.  Remarks on nonlinear parabolic equations. , 1965 .

[6]  J. Vázquez,et al.  On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term , 1995 .

[7]  A. Galaktionov,et al.  Incomplete blow-up and singular interfaces for quasilinear heat equations , 1997 .

[8]  Y. Giga,et al.  Asymptotically self‐similar blow‐up of semilinear heat equations , 1985 .

[9]  Yoshikazu Giga,et al.  Characterizing Blow-up Using Similarity Variables , 1985 .

[10]  Andrew Alfred Lacey,et al.  Global, Unbounded Solutions to a Parabolic Equation , 1993 .

[11]  R. Hamilton.  Three-manifolds with positive Ricci curvature , 1982 .

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

[13]  F. W. Warner,et al.  Curvature Functions for Compact 2-Manifolds , 1974 .

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

[15]  Victor A. Galaktionov,et al.  The problem of blow-up in nonlinear parabolic equations , 2002 .

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

[17]  Juan Luis Vázquez,et al.  Domain of existence and blowup for the exponential reaction-diffusion equation , 1999 .

[18]  Kyûya Masuda.  Analytic solutions of some nonlinear diffusion equations , 1984 .

[19]  Geometrization of 3-Manifolds via the Ricci Flow , 2003 .

[20]  A. K. Kapila.  Review: Jerrold Bebernes and David Eberly, Mathematical problems from combustion theory , 1990 .

[21]  R. Hamilton,et al.  The formations of singularities in the Ricci Flow , 1993 .

[22]  J. J. L. Velázquez,et al.  Classification of singularities for blowing up solutions in higher dimensions , 1993 .

[23]  J. Vázquez,et al.  Theory of Extended Solutions¶for Fast-Diffusion Equations¶in Optimal Classes of Data.¶Radiation from Singularities , 2002 .

[24]  Yi Li,et al.  Travelling Fronts in Cylinders and Their Stability , 1997 .

[25]  Hermann Brunner,et al.  Blowup in diffusion equations: a survey , 1998 .

[26]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[27]  Howard A. Levine,et al.  Quenching, nonquenching, and beyond quenching for solution of some parabolic equations , 1989 .

[28]  Julio D. Rossi,et al.  Thermal avalanche for blowup solutions of semilinear heat equations , 2004 .

[29]  J. Vázquez,et al.  Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations , 1995 .

[30]  M. Fila,et al.  Connecting equilibria by blow-up solutions , 1999 .

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