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SIAM J. on Mathematical Analysis

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1998

Volume 29, Issue 6, pp. 1301-1536


Blow-up in Nonlocal Reaction-Diffusion Equations

Philippe Souplet

SIAM J. Math. Anal. 29, pp. 1301-1334 (34 pages) | Cited 40 times

Online Publication Date: August 01, 2006

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We present new blow-up results for reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of the unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. For each type of problems, we give finite time blow-up results which significantly improve or extend previous results of several authors. In some cases, when the nonlocal source term is in competition with a local dissipative or convective term, optimal conditions on the parameters for finite time blow-up or global existence are obtained.
Our proofs rely on comparison techniques and on a variant of the eigenfunction method combined with new properties on systems of differential inequalities. Moreover, a unified local existence theory for general nonlocal semilinear parabolic equations is developed.

Adiabatic Invariant of the Harmonic Oscillator, Complex Matching and Resurgence

Carles Bonet, David Sauzin, Tere Seara, and Marta València

SIAM J. Math. Anal. 29, pp. 1335-1360 (26 pages) | Cited 3 times

Online Publication Date: August 01, 2006

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The linear oscillator equation with a frequency slowly dependent on time is used to test a method to compute exponentially small quantities. This work presents the matching method in the complex plane as a tool to obtain rigorously the asymptotic variation of the action of the associated Hamiltonian beyond all orders.
The solution in the complex plane is approximated by a series in which all terms present a singularity at the same point. Following matching techniques near this singularity one is led to an equation which does not depend on any parameter, the so-called inner equation, of a Riccati-type. This equation is studied by resurgence methods.

Localized Spatial Homogenization and Large Diffusion

Aníbal Rodríguez-Bernal

SIAM J. Math. Anal. 29, pp. 1361-1380 (20 pages) | Cited 3 times

Online Publication Date: August 01, 2006

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We analyze singular perturbations in elliptic equations, subjected to various boundary conditions, in which the diffusion is going to infinity in localized regions inside the domain and therefore solutions undergo a localized spatial homogenization. The limiting elliptic operator is analyzed as well as convergence of solutions, eigenvalues, and eigenvectors.

Finite Speed of Propagation for the Porous Media Equation

S. Bonafede, G. R. Cirmi, and A. F. Tedeev

SIAM J. Math. Anal. 29, pp. 1381-1398 (18 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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We consider the Cauchy--Dirichlet problem for the equation \[ u_{t} = \Delta u^{m}, \quad m>1, \quad {\rm on} \ D=\Re^{N}_{k} \times ( t>0), \] where $\Re^{N}_{k} = \Re^{N} \cap \left\{x_{1},...,x_{k}>0 \right\}, \ 1 \leq k \leq N, \ N \geq 1$. Sharp bounds of the interface (or free boundary) are obtained. We use a weighted energy method, which allows us to consider more general equations.

Bifurcation to Spiral Waves in Reaction-Diffusion Systems

Arnd Scheel

SIAM J. Math. Anal. 29, pp. 1399-1418 (20 pages) | Cited 8 times

Online Publication Date: August 01, 2006

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For a large class of reaction-diffusion systems on the plane, we show rigorously that m-armed spiral waves bifurcate from a homogeneous equilibrium when the latter undergoes a Hopf bifurcation. In particular, we construct a finite-dimensional manifold which contains the set of small rotating waves close to the homogeneous equilibrium. Examining the flow on this center-manifold in a very general example, we find different types of spiral waves, distinguished by their speed of rotation and their asymptotic shape at large distances of the tip. The relation to the special class of $\lambda$-$\omega$ systems and the validity of these systems as an approximation is discussed.

The Surface Diffusion Flow for Immersed Hypersurfaces

Joachim Escher, Uwe F. Mayer, and Gieri Simonett

SIAM J. Math. Anal. 29, pp. 1419-1433 (15 pages) | Cited 14 times

Online Publication Date: August 01, 2006

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We show existence and uniqueness of classical solutions for the motion of immersed hypersurfaces driven by surface diffusion. If the initial surface is embedded and close to a sphere, we prove that the solution exists globally and converges exponentially fast to a sphere. Furthermore, we provide numerical simulations showing the creation of singularities for immersed curves.

Blowup and Life Span of Solutions for a Semilinear Parabolic Equation

Noriko Mizoguchi and Eiji Yanagida

SIAM J. Math. Anal. 29, pp. 1434-1446 (13 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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This paper is concerned with the Cauchy problem \[ \left \{ \begin{array}{ll} u _t = \Delta u + |u| ^{ p-1 } u & \quad \mbox{ in } \mathbf{R}^N \times (0, \infty), \\ u (x,0) = u_0(x) & \quad \mbox{ in } \mathbf{R}^N, \end{array} \right. \] where $ p > 1 $. Let $\Omega$ be a set in $\mathbf{R}^N$ given by \[ \Omega \equiv \left\{ (r,\omega) \in \mathbf{R}^+ \times S^{N-1} \ : \ r > R, \ d ( \omega,\omega_0 ) < c r ^{-\mu} \right\} \] for some $R > 0$, $c > 0$, $\omega_0 \in S^{N-1}$, and $0 \le \mu < 1$, where $ d(\cdot,\cdot)$ denotes the standard distance on $ S^{N-1}$. It is shown that if $ u_0 $ decays like $ |x| ^{- \alpha }$ as $ |x| \rightarrow \infty $ in $ \Omega$ with $ 0 < \alpha < 2(1-\mu)/(p-1) $, then the solution blows up in finite time regardless of the behavior of $u_0$ outside $\Omega$. Moreover the life span of such a solution with $u_0 = \lambda \varphi$ is estimated from above for small $\lambda > 0$ in terms of $p$, $ \alpha $, and $ \mu $. The optimality of these results is also studied.

Interior Blowup in a Convection-Diffusion Equation

Christopher P. Grant

SIAM J. Math. Anal. 29, pp. 1447-1458 (12 pages)

Online Publication Date: August 01, 2006

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This paper addresses the qualitative behavior of a nonlinear convection-diffusion equation on a smooth bounded domain in Rn, in which the strength of the convection grows superlinearly as the density increases. While the initial-boundary value problem is guaranteed to have a local-in-time solution for smooth initial data, it is possible for this solution to be extinguished in finite time. We demonstrate that the way this may occur is through finite-time "blow up," i.e., the unboundedness of the solution in arbitrarily small neighborhoods of one or more points in the closure of the spatial domain. In special circumstances, such as the presence of radial symmetry, the set of blowup points can be identified; these points may be either on the boundary or on the interior of the domain. Furthermore, criteria can be established that guarantee that blowup occurs. In this paper, such criteria are presented, involving the dimension of the space, the growth rate of the nonlinearity, the strength of the imposed convection field, the diameter of the domain, and the mass of the initial data. Furthermore, the temporal rate of blowup is estimated.

Inequalities for the Polygamma Functions

Horst Alzer and Jim Wells

SIAM J. Math. Anal. 29, pp. 1459-1466 (8 pages) | Cited 5 times

Online Publication Date: August 01, 2006

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Let F_n(x; c) = (\Psi^{(n)}(x))^2 - c \Psi^{(n - 1)}(x) \Psi^{(n + 1)}(x) \quad (x > 0), where $\Psi$ denotes the logarithmic derivative of the gamma function, $n \geq 2$ is an integer, and $c$ is a real number. The authors prove that the function $x \mapsto F_n (x; \alpha)$ is strictly completely monotonic on $(0, \infty)$ if and only if $\alpha \leq (n - 1)/n$, while $x \mapsto - F_n(x; \beta)$ is strictly completely monotonic on $(0, \infty)$ if and only if $\beta \geq n/(n + 1)$.

A Method of Characteristics for Some Systems of Conservation Laws

Arnaud Heibig and Amina Sahel

SIAM J. Math. Anal. 29, pp. 1467-1480 (14 pages)

Online Publication Date: August 01, 2006

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We study systems of conservation laws which belong to the Temple class. Some algebraic formulas are derived and used to integrate the Cauchy problem. In particular, the method of characteristics is extended to the case of a system of two coupled equations.

On the Dirichlet Problem for Vectorial Hamilton--Jacobi Equations

Sandro Zagatti

SIAM J. Math. Anal. 29, pp. 1481-1491 (11 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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We give sufficient conditions for the existence of solutions to the Hamilton--Jacobi equations with Dirichlet boundary condition: $$ \cases{ g(x,{\hbox{\rm det}}Du(x))=0, \ & for a.e. $x\in\Omega,$\cr u(x)=\varphi(x), & for $x\in\partial\Omega,$} $$ obtaining, in addition, an application to the theory of existence of minimizers for a class of nonconvex variational problems.

On the Stationary Cahn--Hilliard Equation: Bubble Solutions

Juncheng Wei and Matthias Winter

SIAM J. Math. Anal. 29, pp. 1492-1518 (27 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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We study stationary solutions of the Cahn--Hilliard equation in a bounded smooth domain that have an interior spherical interface (bubbles). We show that a large class of interior points (the "nondegenerate peak" points) have the following property: there exists such a solution whose bubble center lies close to a given nondegenerate peak point. Our construction uses, among others, the Liapunov--Schmidt reduction method and exponential asymptotics.

Singular Perturbation Approach to a 3-component Reaction-Diffusion System Arising in Population Dynamics

Yukio Kan-on and Masayasu Mimura

SIAM J. Math. Anal. 29, pp. 1519-1536 (18 pages) | Cited 12 times

Online Publication Date: August 01, 2006

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In order to understand theoretically predation-mediated coexistence of competing species which is often observed in ecological systems, we consider a 3-component reaction-diffusion system describing the interaction of one predator and two competing prey species which move by diffusion. It is shown that there exist stable spatially inhomogeneous positive equilibrium solutions of the one-dimensional system under the Neumann boundary condition. This implies ecologically that in the presence of the predator, two competing species coexist with spatially segregating structures. The main tools we use are the singular perturbation technique and the associated singular limit spectral analysis.
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