The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations

References

• Abdellaoui , B and Peral Alonso , I . 2004 . Holder regularity and Harnack inequality for degenerate parabolic equations related to Caffarelli–Kohn–Nirenberg inequalities . Nonlinear Analysis , 57 : 971 – 1003 .
   Web of Science ®Google Scholar
• Andreucci , D and Di Benedetto , E . 1990 . A new approach to initial traces in nonlinear filtration . Annales De L Institut Henri Poincare , : 305 – 334 .
   Google Scholar
• Andreucci , D , Cirmi , GR , Leonardi , S and Tedeev , AF . 2001 . Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary . Journal of Differential Equations , 174 : 253 – 288 .
   Google Scholar
• Andreucci , D and Tedeev , AF . 1999 . A Fujita type result for degenerate Neumann problem in domains with noncompact boundary . Journal of Mathematical Analysis and Applications , 231 : 543 – 567 .
   Web of Science ®Google Scholar
• Andreucci , D and Tedeev , AF . 2000 . Sharp estimates and finite speed of propagation for a Neumann problem in domains narrowing at infinity . Advances in Differential Equations , 5 : 833 – 860 .
   Google Scholar
• Andreucci , D and Tedeev , AF . 2001 . Finite speed of propagation for thin film equations and other higher order parabolic equations with general nonlinearity . Interfaces and Free Boundaries , 3 ( 3 ) : 233 – 264 .
   Google Scholar
• Andreucci , D and Tedeev , AF . 2005 . Universal bounds at the blow-up time for nonlinear parabolic equations . Advances in Differential Equations , 10 ( 1 ) : 89 – 120 .
   Web of Science ®Google Scholar
• Antoncev , SN . 1981 . On the localization of solutions of nonlinear degenerate elliptic and parabolic equations . Soviet Mathematics Docklady , 24 : 420 – 424 .
   Google Scholar
• Antontsev , SN , Diaz , JI and Shmarev . 2002 . “ Energy methods for the free boundary problems ” . In Applications to nonlinear PDEs and fluid mechanics , Boston : Birkhauser .
   Google Scholar
• Benilan , PhCrandall , MG and Pierre , M . 1984 . Solutions of the porous medium equation in R^N under optimal conditions on initial values . Indiana University Mathematics Journal , 33 : 51 – 87 .
   Web of Science ®Google Scholar
• Bernis , F . 1988 . Qualitative properties for some nonlinear higher order degenerate parabolic equations . Houston Journal of Mathematics , 14 ( 3 ) : 319 – 352 .
   Web of Science ®Google Scholar
• Bernis , F . 1988 . Existence results for doubly nonlinear higher order parabolic equations on unbounded domains . Mathematische Annalen , 279 : 373 – 394 .
   Web of Science ®Google Scholar
• Bonafede , S and Skrypnik , II . 1999 . On holder continuity of solutions of doubly nonlinear parabolic equations with weight . Ukrainian Mathematical Journal , 51 ( 7 ) : 890 – 903 .
   Google Scholar
• Caffarelli , L , Kohn , R and Nirenberg , L . 1984 . First order interpolation inequalities with weights . Composito Mathematica , 53 : 259 – 275 .
   Web of Science ®Google Scholar
• Diaz , J and Veron , L . 1985 . Local vanishing properties of solutions of elliptic and parabolic quasilinear equations . Transactions of the American Mathematical Society , 290 ( 2 ) : 787 – 814 .
   Web of Science ®Google Scholar
• Di Benedetto , E and Herrero , MA . 1989 . On the Cauchy problem and initial traces for a degenerate parabolic equation . Transactions of the American Mathematical Society , 314 ( 1 ) : 187 – 224 .
   Web of Science ®Google Scholar
• Di Benedetto , E and Kwong , YC . 1989 . Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations . Transactions of the American Mathematical Society , 330 ( 2 ) : 783 – 811 .
   Web of Science ®Google Scholar
• Eidus , D and Kamin , S . 1994 . The filtration equation in class of functions decreasing at infinity . Proceedings of the American Mathematical Society , 120 ( 3 ) : 825 – 830 .
   Web of Science ®Google Scholar
• Galaktionov , V.A and King , JR . 1996 . On the behaviour of blow-up interfaces for a inhomogeneous filtration equation . Journal of Applied Mathematics , 57 : 53 – 77 .
   Web of Science ®Google Scholar
• Galaktionov , VA , Kamin , S , Kersner , R and Vazquez , JL . 2004 . Intermediate asymptotics for inhomogeneous nonlinear heat conduction . Journal of Mathematical Sciences , 120 ( 3 ) : 1277 – 1294 .
   Google Scholar
• Guedda , M , Hihorst , D and Peletier , MA . 1997 . Disappearing interfaces in nonlinear diffusion . Advances in Mathematical Sciences and Applications , 7 : 695 – 710 .
   Google Scholar
• Ishige , K . 1996 . On the existence of solutions of the Cauchy problem for a doubly nonlinear parabolic equations . SIAM Journal on Mathematical Analysis , 27 ( 5 ) : 1235 – 1260 .
   Web of Science ®Google Scholar
• Kalashnikov , AS . 1987 . Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations . Uspekhi Mathematicheskaya Nauk , 42 : 135 – 176 . English translation in: Russian Mathematical Surveys, 42169–222.
   Google Scholar
• Kamin , S . 1996 . “ Heat propagation in an inhomogeneous medium ” . In Progress in Partial Differential Equations: The Metz Surveys 4 Pitman Research Notes Math , 345 229 – 237 . Harlow : Longman .
   Google Scholar
• Kamin , S and Kersner , R . 1993 . Disappearance of interfaces in finite time . Mechanica , 28 : 117 – 120 .
   Google Scholar
• Kamin , S , Kersner , R and Tesei , A . 1998 . On the Cauchy problem for a class of parabolic equations with variable density . Rendicontidella Mathematica AccademiaLincei , 9 ( 9 ) : 279 – 298 .
   Google Scholar
• Kamin , S and Rosenau , P . 1981 . Propagation of thermal waves in an inhomogeneous medium . Communications on Pure and Applied Mathematics , 34 : 831 – 852 .
   Web of Science ®Google Scholar
• Kamin , S and Rosenau , P . 1982 . Nonlinear diffusion in finite mass medium . Communications on Pure Applied Mathematics , 35 : 113 – 127 .
   Web of Science ®Google Scholar
• Kamin , S and Vazquez , JL . 1988 . Fundamental solutions and asymptotic behaviour for the p-Laplacian equation . Revista Matematica Iberoamericana , 4 ( 2 ) : 339 – 354 .
   Google Scholar
• Kersner , R , Reyes , G and Tesei , A . 2002 . On a class of parabolic equations with variable density and absorption . Advances in Differential Equations , 7 ( 2 ) : 155 – 176 .
   Google Scholar
• Ladyzhenskaya , OA , Solonnikov , VA and Ural'ceva , NN . 1968 . Linear and Quasilinear Equations of Parabolic Type23 of Translation of Mathematical MonographsAmerican Mathematical SocietyProvidenceRI. ,
   Google Scholar
• Martynenko , AV and Tedeev , AF . On the behaviour of solutions to the Cauchy problem for the degenerate parabolic equation with nonhomogenous density and sources (To appear in Computational Mathematics and Mathematical Physics) ,
   Google Scholar
• Maz'ya , V . 1985 . Sobolev spaces , New York : Springer-Verlag .
   Google Scholar
• Reyes , G and Vazquez , JL . 2006 . The Cauchy problem for the inhomogeneous porous medium equation. . Networks and Heterogeneous Media , 1 ( 2 ) : 337 – 351 .
   Web of Science ®Google Scholar
• Skrypnik , II and Tedeev , AF . 1997 . Local estimates of a Cauchy problem for second order parabolic equations. Weighted case 1 . Siberian Mathematical Journal , 38 : 165 – 178 .
   Google Scholar
• Tedeev , AF . 2004 . Conditions for the time global existence and nonexistence of a compact support of solutions to the Cauchy problem for quasilinear parabolic equations . Siberian Mathematical Journal , 45 ( 1 ) : 155 – 164 .
   Web of Science ®Google Scholar
• Tsutsumi , M . 1988 . On solutions of some doubly nonlinear degenerate parabolic equations with absorption, . Journal of Mathematical Analysis and Applications , 132 : 187 – 212 .
   Web of Science ®Google Scholar