zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Some results on the stability and bifurcation of stationary solutions of delay-diffusion equations. (English) Zbl 0881.35120
Nonlinear delay-diffusion equations of the form $$u_t= u_{xx}(x,t)+ a(x)u(x,t)+ bu(x,t-\tau)+ g(x,u(x,t))$$ are studied and stability and bifurcation results are obtained by studying the perturbation of the spectrum of the linearization $$u_t= u_{xx}(x,t)+ a(x)u(x,t)+ bu(x,t-\tau)$$ caused by $g$. It is shown that a centre manifold of any dimension exists for some $a$, $b$, and $\tau$. Moreover, it is shown that if $\pm i\omega_0,\pm i\omega_1,\dots,\pm i\omega_{m-1}$ are the $2m$ pure imaginary eigenvalues of the linearized equation, then the numbers $\omega_0,\omega_1,\dots,\omega_{m-1}$ are rationally independent and as $\tau$ increases through the critical value, the nonlinear equation undergoes a bifurcation from $2\ell$ eigenvalues, where $\ell$ is the sum of the multiplicities of the $m$ eigenvalues.

MSC:
35R10Partial functional-differential equations
35K57Reaction-diffusion equations
WorldCat.org
Full Text: DOI
References:
[1] Casten, R.; Holland, C.: Instability results for reaction-diffusion equations with Neumann boundary conditions. J. differential equations 27, 266-273 (1978) · Zbl 0338.35055
[2] Chafee, N.: The electric balast resistor: homogeneous and nonhomogeneous equilibria. Nonlinear differential equations: invariance, stability and bifurcation, 97-127 (1981)
[3] Desch, W.; Schappacher, W.: Linearized stability for nonlinear semigroups. Lecture notes in math. 1223 (1986) · Zbl 0615.47048
[4] Farmer, J. D.: Chaotic attractors of an infinite-dimensional dynamical system. Physica D 4, 366-393 (1982) · Zbl 1194.37052
[5] Freitas, P.: Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations. J. dynamics differential equations 6, 613-629 (1994) · Zbl 0807.35068
[6] P. Freitas, M. Vishnevskii, Stability of stationary solutions of nonlocal reaction-diffusion equations inm, Differential Integral Equations · Zbl 1038.35030
[7] Friesecke, G.: Convergence to equilibrium for delay-diffusion equations with small delay. J. dynamics differential equations 5, 89-103 (1993) · Zbl 0798.35150
[8] Hale, J. K.: Flows on centre manifolds for scalar functional differential equations. Proc. roy. Soc. Edinburgh sect. 101, 193-201 (1985) · Zbl 0582.34058
[9] Hale, J. K.; Lin, X. -B.: Symbolic dynamics and nonlinear flows. Ann. mat. Pura appl. 144, 229-259 (1986) · Zbl 0632.58027
[10] Hale, J. K.; Sternberg, N.: Onset of chaos in differential delay equations. J. comput. Phys. 77, 221-239 (1988) · Zbl 0644.65050
[11] C.-S. Lin, W.-M. Ni, On stable states of semilinear diffusion equations
[12] Lin, X.; So, J. W. -H.; Wu, J.: Centre manifolds for partial differential equations with delays. Proc. roy. Soc. Edinburgh sect. 122, 237-254 (1992) · Zbl 0801.35062
[13] Matano, H.: Asymptotic behaviour and stability of solutions of semilinear diffusion equations. Publ. res. Inst. math. Sci. 15, 401-454 (1979) · Zbl 0445.35063
[14] Martin, R. H.; Smith, H. L.: Reaction-diffusion systems with time delays: monoticity, invariance, comparison, and convergence. J. reine angew. Math. 413, 1-35 (1991) · Zbl 0709.35059
[15] Parrott, M. E.: Linearized stability and irreducibility for a functional differential equation. SIAM J. Math. anal. 23, 649-661 (1992) · Zbl 0763.34058
[16] Pöschel, J.; Trubowitz, E.: Inverse spectral theory. Pure and applied mathematics 130 (1987)
[17] Rybakowski, K. P.: Realization of arbitrary vector fields on invariant manifolds of delay equations. J. differential equations 144, 222-231 (1994) · Zbl 0815.34064
[18] Sweers, G.: Semilinear elliptic problems on domains with corners. Comm. partial differential equations 14, 1229-1247 (1989) · Zbl 0702.35097
[19] Travis, C. C.; Webb, G. F.: Existence and stability for partial functional differential equations. Trans. amer. Math. soc. 200, 395-418 (1974) · Zbl 0299.35085
[20] Walter, H. -O.: Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations. Mem. the amer. Math. soc. 79 (1989)