zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global existence and blow-up problems for reaction diffusion model with multiple nonlinearities. (English) Zbl 1140.35010
Summary: We study the following reaction diffusion model, $$u_t= \nabla\cdot(a(u) b(x) c(t)\nabla u)+ g(x,t) f(u)\qquad\text{in }D\times (0,T),$$ $${\partial u\over\partial n}= h(x, t)r(u)\qquad\text{on }\partial D\times (0,T),$$ $$u(x,0)= u_0(x)> 0\qquad\text{in }\overline D,$$ where $D$ is a bounded domain $\bbfR^N$ with smooth boundary $\partial D$, $N\ge 2$. This paper deals with interactions among three kinds of nonlinear mechanisms: nonlinear reaction, nonlinear convection and nonlinear boundary flux. The existence theorems of blow-up positive solutions, upper bound of “blow-up time”, upper estimates of “blow-up rate”, existence theorems of global positive solutions, and upper estimates of global positive solutions are given under suitable assumptions on $a$, $b$, $c$, $f$, $g$, $h$, $r$ and initial data $u_0(x)$.

35K60Nonlinear initial value problems for linear parabolic equations
35K57Reaction-diffusion equations
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
Full Text: DOI
[1] Acosta, G.; Rossi, J. D.: Blow-up vs. Global existence for quasilinear parabolic systems with a nonlinear boundary condition, Z. angew. Math. phys. 48, 711-724 (1997) · Zbl 0893.35046 · doi:10.1007/s000330050060
[2] Amann, H.: Parabolic evolutions equations and nonlinear boundary conditions, J. differential equations 72, 201-269 (1988) · Zbl 0658.34011 · doi:10.1016/0022-0396(88)90156-8
[3] Chen, S. H.; Yu, D. M.: Global existence and blowup solutions for quasilinear parabolic equations, J. math. Anal. appl. 335, No. 1, 151-167 (2007) · Zbl 1138.35047 · doi:10.1016/j.jmaa.2007.01.066
[4] Chlebik, M.; Fila, M.: On the blow up rate for the heat equation with a nonlinear boundary condition, Math. methods appl. Sci. 23, 1323-1330 (2000) · Zbl 0980.35073 · doi:10.1002/1099-1476(200010)23:15<1323::AID-MMA167>3.0.CO;2-W
[5] Ding, J. T.; Li, S. J.: Blow-up solutions and global solutions for a class of quasilinear parabolic equations with Robin boundary conditions, Comput. math. Appl. 49, 689-701 (2005) · Zbl 1078.35057 · doi:10.1016/j.camwa.2004.11.006
[6] Fila, M.; Quittner, P.: The blow up rate for the heat equation with a nonlinear boundary condition, Math. methods appl. Sci. 14, 197-205 (1991) · Zbl 0735.35014 · doi:10.1002/mma.1670140304
[7] Hu, B.; Yin, H. M.: The profile near blowup time for solution of the heat equation with a nonlinear boundary condition, Trans. amer. Math. soc. 36, 117-135 (1994) · Zbl 0823.35020 · doi:10.2307/2154944
[8] Levine, H. A.: Stability and instability for solutions of burger’s equation with a nonlinear boundary conditions, SIAM J. Math. anal. 19, 312-336 (1988) · Zbl 0696.35159 · doi:10.1137/0519023
[9] Levine, H. A.; Payne, L. E.: Nonexistence theorems for the heat equation with nonlinear boundary condition and for the porous medium equation backward in time, J. differential equations 16, 319-334 (1974) · Zbl 0285.35035 · doi:10.1016/0022-0396(74)90018-7
[10] Levine, H. A.; Payne, L. E.: Some nonexistence theorems for initial-boundary value problem with nonlinear boundary conditions, Proc. amer. Math. soc. 46, 277-284 (1978) · Zbl 0293.35004 · doi:10.2307/2039910
[11] Pablo, A. D.; Quirós, F.; Rossi, J. D.: Asymptotic simplification for a reaction -- diffusion problem with a nonlinear boundary condition, IMA J. Appl. math. 67, 69-98 (2002) · Zbl 1001.35071 · doi:10.1093/imamat/67.1.69
[12] Pao, C. V.: Quasilinear parabolic and elliptic equations with nonlinear boundary conditions, Nonlinear anal. 66, 639-662 (2007) · Zbl 1105.35053 · doi:10.1016/j.na.2005.12.007
[13] Protter, M. H.; Weinberger, H. F.: Maximum principles in differential equations, (1967) · Zbl 0153.13602
[14] Song, X. F.; Zheng, S. N.: Blow-up and blow-up rate for a reaction -- diffusion model with multiple nonlinearities, Nonlinear anal. 54, 279-289 (2003) · Zbl 1024.35049 · doi:10.1016/S0362-546X(03)00063-4
[15] Sperb, R. P.: Maximum principles and their applications, (1981) · Zbl 0454.35001
[16] Walter, W.: On existence and nonexistence in the large of solutions of parabolic differential equation with nonlinear boundary condition, SIAM J. Math. anal. 24, 85-90 (1975) · Zbl 0268.35052 · doi:10.1137/0506008
[17] Wang, J.; Wang, Z. J.; Yin, J. X.: A class of degenerate diffusion equations with mixed boundary conditions, J. math. Anal. appl. 298, 589-603 (2004) · Zbl 1061.35031 · doi:10.1016/j.jmaa.2004.05.028
[18] Wang, M. X.; Wu, Y. H.: Global existence and blow-up problems for quasilinear parabolic conditions with nonlinear boundary conditions, SIAM J. Math. anal. 24, 1515-1521 (1993) · Zbl 0790.35042 · doi:10.1137/0524085
[19] Wonlanski, N.: Global behaviour of positive solutions to nonlinear diffusion problem with nonlinear absorption through boundary, SIAM J. Math. anal. 24, 317-326 (1993) · Zbl 0778.35047 · doi:10.1137/0524021