×

A priori bounds and complete blow up of positive solutions of indefinite superlinear parabolic problems. (English) Zbl 1071.35026

Summary: We study a priori estimates of positive solutions of the equation \[ \partial_t u-\Delta u=\lambda u+ a(x)u^p,\quad x\in\Omega,\quad t> 0, \] satisfying the homogeneous Dirichlet boundary conditions. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(\lambda\in \mathbb{R}\), \(p> 1\) is subcritical, \(a\in C(\overline\Omega)\) changes sign and \(a\), \(p\) satisfy some additional technical hypotheses. Assume that the solution \(u\) blows up in a finite time \(T\) and the set \(\Omega^+ := \{x\in\Omega: a(x)> 0\}\) is connected. Using our a priori bounds, we show that u blows up completely in \(\Omega^+\) at \(t= T\) and the blow up time \(T\) depends continuously on the initial data.

MSC:

35B45 A priori estimates in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] N. Ackermann, T. Bartsch, Superstable manifolds of nondissipative parabolic problems, J. Dynam. Differential Equations, in press · Zbl 1129.35428
[2] N, Ackermann, T. Bartsch, P. Kaplický, P. Quittner, A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems, preprint · Zbl 1143.37049
[3] Amann, H.; López-Gómez, J., A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. differential equations, 146, 336-374, (1998) · Zbl 0909.35044
[4] H. Amann, P. Quittner, Optimal control problems governed by semilinear parabolic equations with non-monotone nonlinearities and low regularity data, preprint · Zbl 1106.49005
[5] Baras, P.; Cohen, L., Complete blow-up after \(T_{\max}\) for the solution of a semilinear heat equation, J. funct. anal., 71, 142-174, (1987) · Zbl 0653.35037
[6] Baras, P.; Kersner, R., Local and global solvability of a class of semilinear parabolic equations, J. differential equations, 68, 238-252, (1987) · Zbl 0622.35033
[7] Berestycki, H.; Capuzzo-Dolcetta, I.; Nirenberg, L., Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. methods nonlinear anal., 4, 59-78, (1994) · Zbl 0816.35030
[8] Bidaut-Véron, M.-F., Initial blow-up for the solutions of a semilinear parabolic equation with source term, (), 189-198 · Zbl 0914.35055
[9] Boudiba, N.; Pierre, M., Global existence for coupled reaction – diffusion systems, J. math. anal. appl., 250, 1-12, (2000) · Zbl 0963.35077
[10] Cazenave, T.; Lions, P.-L., Solutions globales d’équations de la chaleur semi linéaires, Comm. partial differential equations, 9, 955-978, (1984) · Zbl 0555.35067
[11] Chen, W.; Li, C., Indefinite elliptic problems in a domain, Discrete contin. dynam. systems, 3, 333-340, (1997) · Zbl 0948.35054
[12] Chipot, M.; Quittner, P., Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions, J. dynam. differential equations, 16, 91-138, (2004) · Zbl 1077.35065
[13] Y. Du, S. Li, Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations, preprint, Univ. of New England, 2003
[14] M. Fila, H. Matano, P. Poláčik, Immediate regularization after blow-up, preprint · Zbl 1108.35092
[15] Galaktionov, V.; Vázquez, J.L., Continuation of blow-up solutions of nonlinear heat equations in several space dimensions, Comm. pure appl. math., 50, 1-67, (1997) · Zbl 0874.35057
[16] Giga, Y., A bound for global solutions of semilinear heat equations, Comm. math. phys., 103, 415-421, (1986) · Zbl 0595.35057
[17] Giga, Y.; Matsui, S.; Sasayama, S., Blow up rate for semilinear heat equation with subcritical nonlinearity, Indiana univ. math. J., 53, 483-514, (2004) · Zbl 1058.35096
[18] Hamada, T., Nonexistence of global solutions of parabolic equation in conical domains, Tsukuba J. math., 19, 15-25, (1995) · Zbl 0842.35044
[19] Hu, B., Remarks on the blowup estimate for solutions of the heat equation with a nonlinear boundary condition, Differential integral equations, 9, 891-901, (1996) · Zbl 0852.35072
[20] Lacey, A.A.; Tzanetis, D., Complete blow-up for a semilinear diffusion equation with a sufficiently large initial condition, IMA J. appl. math., 41, 207-215, (1988) · Zbl 0699.35151
[21] J. López-Gómez, P. Quittner, Complete and energy blow-up in indefinite superlinear parabolic problem, Discrete Contin. Dynam. Systems, in press
[22] Martel, Y., Complete blow up and global behaviour of solutions of \(u_t - \Delta u = g(u)\), Ann. inst. H. Poincaré anal. non linéaire, 15, 687-723, (1998) · Zbl 0914.35057
[23] Matos, J.; Souplet, Ph., Universal blow-up rates for a semilinear heat equation and applications, Adv. differential equations, 8, 615-639, (2003) · Zbl 1028.35065
[24] Mizoguchi, N., On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity, Math. Z., 239, 215-229, (2002) · Zbl 1008.35033
[25] N. Mizoguchi, Multiple blowup of solutions for a semilinear heat equation, preprint · Zbl 1063.35079
[26] N. Mizoguchi, Multiple blowup of solutions for a semilinear heat equation II, preprint · Zbl 1122.35038
[27] Pierre, M.; Schmitt, D., Blowup in reaction – diffusion systems with dissipation of mass, SIAM rev., 42, 93-106, (2000) · Zbl 0942.35033
[28] Pinsky, R.G., Existence and nonexistence of global solutions for \(u_t = \Delta u + a(x) u^p\) in \(R^d\), J. differential equations, 133, 152-177, (1997) · Zbl 0876.35048
[29] Quirós, F.; Rossi, J.D.; Vázquez, J.L., Thermal avalanche for blowup solutions of semilinear heat equations, Comm. pure appl. math., 57, 59-98, (2004) · Zbl 1036.35084
[30] Quittner, P., A priori bounds for global solutions of a semilinear parabolic problem, Acta math. univ. Comenian., 68, 195-203, (1999) · Zbl 0940.35112
[31] Quittner, P., Continuity of the blow-up time and a priori bounds for solutions in superlinear parabolic problems, Houston J. math., 29, 757-799, (2003) · Zbl 1034.35013
[32] Quittner, P., Multiple equilibria, periodic solutions and a priori bounds for solutions in superlinear parabolic problems, Nodea nonlinear differential equations appl., 11, 237-258, (2004) · Zbl 1058.35120
[33] P. Quittner, Complete and energy blow-up in superlinear parabolic problems, preprint · Zbl 1144.35402
[34] Quittner, P.; Souplet, Ph., A priori estimates of global solutions of superlinear parabolic problems without variational structure, Discrete contin. dynam. systems, 9, 1277-1292, (2003) · Zbl 1029.35049
[35] Ph. Souplet, Optimal regularity conditions for elliptic problems via \(L_\delta^p\) spaces, Duke Math. J., in press
[36] Suzuki, R., Complete blow-up for quasilinear degenerate parabolic equations, Proc. roy. soc. Edinburgh sect. A, 130, 877-908, (2000) · Zbl 0959.35101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.