Publication: Blow-up under oscillatory boundary conditions
Loading...
Full text at PDC
Publication Date
1994
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
IOS Press
Citations
Exportar
Abstract
The object of this paper is the study of blowing-up phenomena for the initial-boundary value problem (Pa): ut=uxx+δeu for (x,t)∈(0,1)×(0,+∞), u(0,t)=asinωt and u(1,t)=0 for t∈[0,+∞), u(x,0)=u0(x) for x∈(0,1), where u0(x) is a continuous and bounded function, and a>0, ω>0 are real constants. It is known that if the amplitude a=0 in the oscillatory boundary condition above then there exists a critical parameter δFK (the so-called Frank-Kamenetskiĭ parameter) such that if δ<δFK the corresponding Cauchy-Dirichlet problem (P0) is globally solvable for suitable choices of u0(x), and each solution of (P0) blows up in a finite time if δ>δFK. The authors prove existence of a parameter δ(a,ω)≤δFK with similar critical properties. The essential part of the paper is devoted to the study of the asymptotic behavior of δ(a,ω) with respect to a and ω. For example, δ(a,ω)∼δFK as a→0 uniformly in ω. Further, the exact dependence of δ(a,ω) on the data in (Pa) is shown in the remaining limiting cases for a and ω.
Description
UCM subjects
Unesco subjects
Keywords
Citation
H. Bellout, A criterion for blow-up of solutions to semilinear heat equations, SIAM J. Math. Anal. 18 (1987) 722-727.
J. Bebernes and D. Eberly, Mathematical Problems from Combustion Theory, Applied Mathematical Sciencies Vol. 83 (Springer-Verlag, Berlin, 1989).
A Friedman, Blow-up of solutions of nonlinear parabolic equations, in: W.M. Ni et al., eds., Nonlinear Diffusion Equations and Their Equilibrium States I (Springer-Verlag, Berlin, 1988) 301-318
I.M. Gelfand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Trans. 29 (1963) 295-381.
M.A Herrero and J.J.L. Velázquez, Blow-up behaviour of one-dimensional semilinear parabolic problems, Ann. Inst. Poincaré 10 (2) (1993) 131-189.
M.A Herrero and J.J.L. Velázquez, Blow-up profiles in one-dimensional, semilinear parabolic problems, Comm. PDE 17 (182) (1992) 205-219.
A.A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math. 43 (1983) 1350-1366.
A.A. Lacey, The form of blow up for nonlinear parabolic equations, Proc. Roy. Soc. Edinburgh 98A (1984) 183-202.
A. Liñán and F.A Williams, Theory of ignition of a reactive solid by constant energy flux, Comb. Sci. Tech. J. 3 (1971) 91-98.
A Liñán and F.A Williams, Ignition of a reactive solid exposed to a step in surface temperature, SIAM J. Appl. Math. 36 (1979) 587-603.