Geodesic
Blow-up of sign-changing solutions to quasilinear parabolic equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 215-224.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the blow-up of sign-changing solutions to the Cauchy problem for quasilinear parabolic equations of arbitrary order. Our approach is based on H. Levine's remarkable idea of constructing a concavity inequality for a negative power of a standard positive definite functional. Combining this with the nonlinear capacity method, which is based on the choice of optimal test functions, we find conditions for the blow-up of solutions to the problems under consideration. 
@article{TM_2010_269_a17,
     author = {S. I. Pohozaev},
     title = {Blow-up of sign-changing solutions to quasilinear parabolic equations},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {215--224},
     publisher = {mathdoc},
     volume = {269},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2010_269_a17/}
}
TY  - JOUR
AU  - S. I. Pohozaev
TI  - Blow-up of sign-changing solutions to quasilinear parabolic equations
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2010
SP  - 215
EP  - 224
VL  - 269
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2010_269_a17/
LA  - ru
ID  - TM_2010_269_a17
ER  - 
%0 Journal Article
%A S. I. Pohozaev
%T Blow-up of sign-changing solutions to quasilinear parabolic equations
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2010
%P 215-224
%V 269
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2010_269_a17/
%G ru
%F TM_2010_269_a17
S. I. Pohozaev. Blow-up of sign-changing solutions to quasilinear parabolic equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function theory and differential equations, Tome 269 (2010), pp. 215-224. http://geodesic.mathdoc.fr/item/TM_2010_269_a17/

[1] Knops R.J., Levine H.A., Payne L.E., “Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics”, Arch. Rat. Mech. Anal., 55 (1974), 52–72 | DOI | MR | Zbl

[2] Levine H.A., “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_{t}=Au+\mathcal F(u)$”, Arch. Rat. Mech. Anal., 51 (1973), 371–386 | DOI | MR | Zbl

[3] Levine H.A., “Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=Au+\mathcal F(u)$”, Trans. Amer. Math. Soc., 192 (1974), 1–21 | MR | Zbl

[4] Levine H.A., “Some additional remarks on the nonexistence of global solutions to nonlinear wave equations”, SIAM J. Math. Anal., 5 (1974), 138–146 | DOI | MR | Zbl

[5] Levine H.A., “Nonexistence of global weak solutions to nonlinear wave equations”, Improperly posed boundary value problems, Res. Notes Math., 1, Pitman, London, 1975, 94–104 | MR

[6] Levine H.A., Pucci P., Serrin J., “Some remarks on global nonexistence for nonautonomous abstract evolution equations”, Harmonic analysis and nonlinear differential equations, Contemp. Math., 208, Amer. Math. Soc., Providence (RI), 1997, 253–263 | DOI | MR | Zbl

[7] Levine H.A., Serrin J., “Global nonexistence theorems for quasilinear evolution equations with dissipation”, Arch. Rat. Mech. Anal., 137 (1997), 341–361 | DOI | MR | Zbl

[8] Levine H.A., Smith R.A., “A potential well theory for the wave equation with a nonlinear boundary condition”, J. reine und angew. Math., 374 (1987), 1–23 | MR | Zbl

[9] Ono K., “Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings”, J. Diff. Equat., 137 (1997), 273–301 | DOI | MR | Zbl

[10] Pucci P., Serrin J., “Asymptotic stability for nonautonomous dissipative wave systems”, Commun. Pure and Appl. Math., 49 (1996), 177–216 | 3.0.CO;2-B class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[11] Pucci P., Serrin J., “Local asymptotic stability for dissipative wave systems”, Israel J. Math., 104 (1998), 29–50 | DOI | MR | Zbl

[12] Pucci P., Serrin J., “Global nonexistence for abstract evolution equations with positive initial energy”, J. Diff. Equat., 150 (1998), 203–214 | DOI | MR | Zbl

[13] Mitidieri E., Pokhozhaev S.I., Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh, Tr. MIAN, 234, Nauka, M., 2001 | Zbl