On the equation in planar domains
Gershon Wolansky (2004)
Banach Center Publications
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The blow-up of solutions for a parabolic equation with nonlocal exponential nonlinearity is studied.
Gershon Wolansky (2004)
Banach Center Publications
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The blow-up of solutions for a parabolic equation with nonlocal exponential nonlinearity is studied.
Aya Khaldi, Amar Ouaoua, Messaoud Maouni (2022)
Mathematica Bohemica
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We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms We prove with suitable assumptions on the variable exponents the global existence of the solution and a stability result using potential and Nihari’s functionals with small positive initial energy, the stability being based on Komornik’s inequality.
Peter Chalov, Vyacheslav Zakharyuta (2011)
Studia Mathematica
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It is proved, using so-called multirectangular invariants, that the condition αβ = α̃β̃ is sufficient for the isomorphism of the spaces and . This solves a problem posed in [14, 15, 1]. Notice that the necessity has been proved earlier in [14].
Fernando Farroni, Raffaella Giova (2011)
Studia Mathematica
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We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk onto itself preserves the space EXP() of exponentially integrable functions over , in the sense that u ∈ EXP() if and only if . Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate for every u ∈ EXP(). Similarly, we consider the distance from in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then for every u ∈ EXP(). We also...
Ph. Souplet (2009)
Journal of the European Mathematical Society
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We consider positive solutions of the system ; in a ball or in the whole space, with . Relatively little is known on the blow-up set for semilinear parabolic systems and, up to now, no result was available for this basic system except for the very special case . Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for...
Jakub Slavík (2021)
Czechoslovak Mathematical Journal
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We study the asymptotic behaviour of solutions of a reaction-diffusion equation in the whole space driven by a spatially homogeneous Wiener process with finite spectral measure. The existence of a random attractor is established for initial data in suitable weighted -space in any dimension, which complements the result from P. W. Bates, K. Lu, and B. Wang (2013). Asymptotic compactness is obtained using elements of the method of short trajectories.
Jan Büthe (2015)
Acta Arithmetica
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The function is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before exp(495.702833165).
Stéphane Descombes, Mohand Moussaoui (2000)
Bollettino dell'Unione Matematica Italiana
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Si considerano equazioni di Ginzburg-Landau complesse del tipo in dove è polinomio di grado a coefficienti complessi e è un numero complesso con parte reale positiva . Nell'ipotesi che la parte reale del coefficiente del termine di grado massimo sia positiva, si dimostra l'esistenza e la regolarità di una soluzione globale nel caso , dove dipende da e .
José M. Arrieta, Anibal Rodriguez-Bernal, Philippe Souplet (2004)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions : either the space derivative blows up in finite time (with itself remaining bounded), or is global and converges in norm to the unique steady state. The main difficulty is to prove boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out...
Noureddine Igbida, Mokhtar Kirane (2002)
Colloquium Mathematicae
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This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form , , , for , t > 0, p > 0, q > 0, r > 0, , under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for , where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the...
Hans-Goerg Roos (2018)
Applications of Mathematics
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Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss the difficulties which arise for systems of reaction-diffusion problems.
Kaňa, Radek, Matonoha, Ctirad, Papáček, Štěpán, Soukup, Jindřich
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We present the method for determination of phycobilisomes diffusivity (diffusion coefficient ) on thylakoid membrane from fluorescence recovery after photobleaching (FRAP) experiments. This was usually done by analytical models consisting mainly of a simple curve fitting procedure. However, analytical models need some unrealistic conditions to be supposed. Our method, based on finite difference approximation of the process governed by the Fickian diffusion equation and on the minimization...
Henrik Petersson (2004)
Studia Mathematica
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Let E,F be Banach spaces where F = E’ or vice versa. If F has the approximation property, then the space of nuclearly entire functions of bounded type, , and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on (i.e. the continuous operators that commute with all translations) is formed by the transposes , φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on is PDE-preserving for a set ℙ ⊆...
Joanna Rencławowicz (2000)
Applicationes Mathematicae
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The Fujita type global existence and blow-up theorems are proved for a reaction-diffusion system of m equations (m>1) in the form with nonnegative, bounded, continuous initial values and positive numbers . The dependence on of the length of existence time (its finiteness or infinitude) is established.