Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations

Joanna Rencławowicz

Displaying similar documents to “Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations”

On the equation $u_{t} = Δ u + M e x p u / \int e x p u d x$ in planar domains

Gershon Wolansky (2004)

Banach Center Publications

Similarity:

The blow-up of solutions for a parabolic equation with nonlocal exponential nonlinearity is studied.

Isomorphic classification of the tensor products $E ₀ (e x p α i) \otimes ̂ E_{\infty} (e x p β j)$

Peter Chalov, Vyacheslav Zakharyuta (2011)

Studia Mathematica

Similarity:

It is proved, using so-called multirectangular invariants, that the condition αβ = α̃β̃ is sufficient for the isomorphism of the spaces $E ₀ (e x p α i) \otimes ̂ E_{\infty} (e x p β j)$ and $E ₀ (e x p α ̃ i) \otimes ̂ E_{\infty} (e x p β ̃ j)$ . This solves a problem posed in [14, 15, 1]. Notice that the necessity has been proved earlier in [14].

Quasiconformal mappings and exponentially integrable functions

Fernando Farroni, Raffaella Giova (2011)

Studia Mathematica

Similarity:

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 $u \circ f^{- 1} \in E X P ()$ . Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate $1 / (1 + K l o g K) \leq (| | u \circ f^{- 1} {| |}_{E X P ()} {) / (| | u | |}_{E X P ()}) \leq 1 + K l o g K$ for every u ∈ EXP(). Similarly, we consider the distance from $L^{\infty}$ in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then $1 / K \leq (d i s t_{E X P (f (G))} (u \circ f^{- 1}, L^{\infty} (f (G)))) / (d i s t_{E X P (f (G))} (u, L^{\infty} (G))) \leq K$ for every u ∈ EXP(). We also...

Single-point blow-up for a semilinear parabolic system

Ph. Souplet (2009)

Journal of the European Mathematical Society

Similarity:

We consider positive solutions of the system $u_{t} - Δ u = v^{p}$ ; $v_{t} - Δ v = u^{q}$ in a ball or in the whole space, with $p, q > 1$ . 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 $p = q$ . 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...

On the first sign change in Mertens' theorem

Jan Büthe (2015)

Acta Arithmetica

Similarity:

The function $\sum_{p \leq x} 1 / p - l o g l o g (x) - M$ 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).

On estimation of diffusion coefficient based on spatio-temporal FRAP images: An inverse ill-posed problem

Kaňa, Radek, Matonoha, Ctirad, Papáček, Štěpán, Soukup, Jindřich

Similarity:

We present the method for determination of phycobilisomes diffusivity (diffusion coefficient $D$ ) 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...

Global existence and regularity of solutions for complex Ginzburg-Landau equations

Stéphane Descombes, Mohand Moussaoui (2000)

Bollettino dell'Unione Matematica Italiana

Similarity:

Si considerano equazioni di Ginzburg-Landau complesse del tipo $u_{t} - α Δ u + P ({|u|}^{2}) u = 0$ in $R^{N}$ dove $P$ è polinomio di grado $K$ a coefficienti complessi e $α$ è un numero complesso con parte reale positiva $ℜ α$ . Nell'ipotesi che la parte reale del coefficiente del termine di grado massimo $P$ sia positiva, si dimostra l'esistenza e la regolarità di una soluzione globale nel caso $|α| < C ℜ α$ , dove $C$ dipende da $K$ e $N$ .

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

José M. Arrieta, Anibal Rodriguez-Bernal, Philippe Souplet (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

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 $u$ : either the space derivative $u_{x}$ blows up in finite time (with $u$ itself remaining bounded), or $u$ is global and converges in $C^{1}$ norm to the unique steady state. The main difficulty is to prove $C^{1}$ boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out...

Blow up for a completely coupled Fujita type reaction-diffusion system

Noureddine Igbida, Mokhtar Kirane (2002)

Colloquium Mathematicae

Similarity:

This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $u ₜ - Δ (a_{11} u) = h (t, x) {| v |}^{p}$ , $v ₜ - Δ (a_{21} u) - Δ (a_{22} v) = k (t, x) {| w |}^{q}$ , $w ₜ - Δ (a_{31} u) - Δ (a_{32} v) - Δ (a_{33} w) = l (t, x) {| u |}^{r}$ , for $x \in ℝ^{N}$ , t > 0, p > 0, q > 0, r > 0, $a_{i j} = a_{i j} (t, x, u, v)$ , under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x \in ℝ^{N}$ , 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...

Rings of PDE-preserving operators on nuclearly entire functions

Henrik Petersson (2004)

Studia Mathematica

Similarity:

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, $ℋ_{N b} (E)$ , and the space of exponential type functions, Exp(F), form a dual pair. The set of convolution operators on $ℋ_{N b} (E)$ (i.e. the continuous operators that commute with all translations) is formed by the transposes $φ (D) \equiv^{t} φ$ , φ ∈ Exp(F), of the multiplication operators φ :ψ ↦ φ ψ on Exp(F). A continuous operator T on $ℋ_{N b} (E)$ is PDE-preserving for a set ℙ ⊆...

Global existence and blow-up for a completely coupled Fujita type system

Joanna Rencławowicz (2000)

Applicationes Mathematicae

Similarity:

The Fujita type global existence and blow-up theorems are proved for a reaction-diffusion system of m equations (m>1) in the form $u_{i t} = Δ u_{i} + u_{i + 1}^{p_{i}}, i = 1, . . ., m - 1,$ $u_{m t} = Δ u_{m} + u_{1}^{p_{m}}, x \in ℝ^{N}, t > 0,$ with nonnegative, bounded, continuous initial values and positive numbers $p_{i}$ . The dependence on $p_{i}$ of the length of existence time (its finiteness or infinitude) is established.

Identification of a localized source in an interstellar cloud: an inverse problem

Meri Lisi, Silvia Totaro (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

We study an inverse problem for photon transport in an interstellar cloud. In particular, we evaluate the position $x_{0}$ of a localized source $q (x) = q_{0} δ (x - x_{0})$ , inside a nebula (for example, a star). We assume that the photon transport phenomenon is one-dimensional. Since a nebula moves slowly in time, the number of photons $U$ inside the cloud changes slowly in time. For this reason, we consider the so-called quasi-static approximation $u$ to the exact solution $U$ . By using semigroup theory, we prove existence...