Nonexistence of global solutions and bifurcation analysis for a boundary-value problem of parabolic type

Author:
C. V. Pao

Journal:
Proc. Amer. Math. Soc. **65** (1977), 245-251

MSC:
Primary 35K60

DOI:
https://doi.org/10.1090/S0002-9939-1977-0454362-7

MathSciNet review:
0454362

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Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is to present a bifurcation analysis on the existence of the nonexistence of a global solution for a semilinear parabolic equation and to characterize the local stability and the instability of the corresponding steady-state solutions. The bifurcation result can be described either by a parameter for a fixed spatial domain or by varying for a fixed . The stability analysis gives a result which can be used to determine the stability or instability problem when the system possesses nonintersecting multiple steady-state solutions.

**[1]**H. Amann,*On the existence of positive solutions of nonlinear elliptic boundary value problems*, Indiana Univ. Math. J.**21**(1971/72), 125-146. MR**45**#5558. MR**0296498 (45:5558)****[2]**C. Bandle,*Existence theorems, qualitative results and a priori bounds for a class of nonlinear Dirichlet problems*, Arch. Rational Mech. Anal.**59**(1975), 219-238. MR**0454336 (56:12587)****[3]**J. P. G. Ewer and L. A. Peletier,*On the asymptotic behavior of solutions of semilinear parabolic equations*, SIAM J. Appl. Math.**28**(1975), 43-53. MR**0407425 (53:11200)****[4]**A. Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Englewood Cliffs, N. J., 1964. MR**31**#6062. MR**0181836 (31:6062)****[5]**H. Fujita,*On the nonlinear equations**and*, Bull. Amer. Math. Soc.**75**(1969), 132-135. MR**39**#615. MR**0239258 (39:615)****[6]**I. M. Gel'fand,*Some problems in the theory of quasilinear equations*, Amer. Math. Soc. Transl. (2)**29**(1963), 295-381. MR**27**#3921. MR**0153960 (27:3921)****[7]**K. Hayakawa,*On non-existence of global solutions of some semi-linear parabolic differential equations*, Proc. Japan Acad.**49**(1973), 503-505. MR**0338569 (49:3333)****[8]**D. D. Joseph and E. M. Sparrow,*Nonlinear diffusion induced by nonlinear sources*, Quart. Appl. Math.**28**(1970), 327-342. MR**42**#7153. MR**0272272 (42:7153)****[9]**H. B. Keller and D. S. Cohen,*Some positone problems suggested by nonlinear heat generation*, J. Math. Mech.**16**(1967), 1361-1376. MR**35**#4552. MR**0213694 (35:4552)****[10]**C. V. Pao,*Successive approximations of some nonlinear initial boundary value problems*, SIAM J. Math. Anal.**5**(1974), 91-102. MR**49**#4265. MR**0339507 (49:4265)****[11]**-,*Positive solutions of a nonlinear boundary value problem of parabolic type*, J. Differential Equations**22**(1976), 145-163. MR**0422876 (54:10862)****[12]**M. H. Protter and H. F. Weinberger,*Maximum principles in differential equations*, Prentice-Hall, Englewood Cliffs, N. J., 1967. MR**36**#2935. MR**0219861 (36:2935)****[13]**-,*On the spectrum of general second order operators*, Bull. Amer. Math. Soc.**72**(1966), 251-255. MR**32**#7939. MR**0190527 (32:7939)****[14]**D. H. Sattinger,*Monotone methods in nonlinear elliptic and parabolic boundary value problems*, Indiana Univ. Math. J.**21**(1971/72), 979-1000. MR**45**#8969. MR**0299921 (45:8969)****[15]**S. Sugitani,*On non-existence of global solutions for some nonlinear integral equations*, Osaka J. Math.**12**(1975), 45-51. MR**0470493 (57:10247)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0454362-7

Keywords:
Nonexistence of global solution,
parabolic equation,
local stability and instability,
bifurcation analysis

Article copyright:
© Copyright 1977
American Mathematical Society