ISSN:

1078-0947

eISSN:

1553-5231

All Issues

## Discrete & Continuous Dynamical Systems - A

2006 , Volume 14 , Issue 1

Special Issue

Qualitative theory of nonlinear elliptic and parabolic problems

**
2006, 14(1): i-iii
doi: 10.3934/dcds.2006.14.1i
**

Interaction between nonlinearity and diffusion is a fascinating subject. Many interesting phenomena are known to result from such interaction, for example, generation of interfaces and singularities, formation of spatial and temporal patterns, propagation of waves, symmetrization and symmetry-breaking of solutions, and so on. Nonlinear diffusive systems have undergone a thorough investigation aimed at mathematical understanding of the mechanisms behind the phenomena.

The analysis serving this aim uses various classical and newly developed techniques relying on results from the bifurcation theory, singular perturbation theory, variational method, and the theory of finite and infinite dimensional dynamical systems. The qualitative theory of parabolic and elliptic equations has been developing extensively in the last few decades, and a lot of important, interesting and beautiful results have been obtained concerning the dynamics of solutions and qualitative description of steady states.

For more information please click the “Full Text” above.
2006, 14(1): 1-29
doi: 10.3934/dcds.2006.14.1

We study the degenerate logistic model described by the equation $ u_t - $Δ$ u=au-b(x)u^p$ with standard boundary conditions, where $p>1$, $b$ vanishes on a nontrivial subset $\Omega_0$ of the underlying bounded domain $\Omega\subset R^N$ and $b$ is positive on $\Omega_+=\Omega\setminus \overline{\Omega}_0$. We consider the difficult case where $\partial\Omega_0\cap \partial \Omega$≠$\emptyset$ and $\partial\Omega_+\cap \partial \Omega$≠$\emptyset$, and examine the asymptotic behaviour of the solutions. By a detailed study of a singularly mixed boundary blow-up problem, we obtain some basic results on the dynamics of the model.
2006, 14(1): 31-62
doi: 10.3934/dcds.2006.14.31

This paper is concerned with the dynamics of travelling spot solutions in two dimensions. Travelling spot solutions are constructed under the bifurcation structure with Jordan block type degeneracy. It is shown that if the velocity is very slow, such travelling spots possess reflection property. In order to do it, we derive the reduced ordinary differential equations describing the dynamics of interacting travelling spots in RD systems by using center manifold theory. This reduction enables us to prove that two very slowly travelling spots reflect before collision as if they were elastic particles.
2006, 14(1): 63-74
doi: 10.3934/dcds.2006.14.63

This paper examines the following question: Suppose that we have a reaction-diffusion equation or system such that some solutions which are homogeneous in space blow up in finite time. Is it possible to inhibit the occurrence of blow-up as a consequence of imposing Dirichlet boundary conditions, or other effects where diffusion plays a role? We give examples of equations and systems where the answer is affirmative.
2006, 14(1): 75-92
doi: 10.3934/dcds.2006.14.75

In this paper we study solutions to reaction-diffusion equations in the bistable case, defined on the whole space in dimension $N$. The existence of solutions with cylindric symmetry is already known. Here we prove the uniqueness of these cylindric solutions whose level sets are curved Lipschitz graphs. Using a centre manifold-like argument, we also give the precise asymptotics of these level sets at infinity. In dimension 2, we classify all solutions under weak conditions at infinity. Finally, we also provide an alternative proof of the existence of these solutions in dimension 2, based on a continuation argument.
2006, 14(1): 93-116
doi: 10.3934/dcds.2006.14.93

We study the existence of multiple positive stable solutions for
2006, 14(1): 117-134
doi: 10.3934/dcds.2006.14.117

The blowup behaviors of solutions to a scalar-field equation with the Robin condition are discussed. For some range of the parameter, there exist at least two positive solutions to the equation. Here, the blowup rate of the large solution and the scaling properties are discussed.
2006, 14(1): 135-148
doi: 10.3934/dcds.2006.14.135

In this paper, we consider a Lotka-Volterra competition model with diffusion, and show that the global bifurcation structure of positive stationary solutions for the model is similar to that for a certain scalar reaction-diffusion equation. To do this, the comparison principle, the bifurcation theory, and the numerical verification are employed.
2006, 14(1): 149-168
doi: 10.3934/dcds.2006.14.149

We study a Ginzburg-Landau energy in a one-dimensional ring with nonuniform thickness, where the nonuniformity is expressed by a piecewise-constant function. That is a simplified model describing a supercurrent in the superconducting ring. Then the Ginzburg-Landau equation with a discontinuous coefficient subject to periodic boundary conditions is derived as the Euler-Lagrange equation of the energy functional. Since the unknown variable of the equation is complex-valued, we can define the phase of a solution if the solution has no zero. The purpose of this article is to establish the existence of nontrivial solutions with no zero and to reveal the configuration of the phase of the solutions as the coefficient converges to zero in a set of subintervals. More precisely we control the convergence of the coefficient with a small positive parameter $\varepsilon$ having various orders in the subintervals and prove the convergence of the solutions to those of a limiting equation as $\varepsilon\to0$ together with the convergence rate. As a consequence, for small $\varepsilon$ most of the phase variation takes place on the subintervals where the coefficient converges to zero with the highest order. Finally we show the stability of those solutions.
2006, 14(1): 169-186
doi: 10.3934/dcds.2006.14.169

We study the continuation of solutions of superlinear indefinite parabolic problems after the blow-up time. The nonlinearity is of the form $a(x)u^p$, where $p>1$ is subcritical and $a$ changes sign. Unlike the case $a>0$, the solutions will never blow up completely in the whole domain but only in a certain subdomain. In some cases we give a precise description of this subdomain. We also derive sufficient conditions for the blow-up of the associated energy.
2006, 14(1): 187-202
doi: 10.3934/dcds.2006.14.187

We consider positive solutions of the equation $- $ε$^2 $Δ $u + u $=$u^p$ in $\Omega$, where $\Omega \subseteq \R^n$, $p > 1$ and ε is a small positive parameter. Neumann boundary conditions are imposed in general. We prove existence of solutions which concentrate at curves or manifolds in $\overline{\Omega}$ when ε → 0.
2006, 14(1): 203-220
doi: 10.3934/dcds.2006.14.203

This paper is concerned with the long time behavior for the evolution of a curve governed by the curvature flow with constant driving force in two-dimensional space. Especially, the asymptotic stability of a traveling wave whose shape is a line is studied. We deal with moving curves represented by the entire graphs on the $x$-axis. By studying the Cauchy problem, the asymptotic stability of traveling waves with spatially decaying initial perturbations and the convergence rate are obtained. Moreover we establish the stability result where initial perturbations do not decay to zero but oscillate at infinity. In this case, we prove that one of the sufficient conditions for asymptotic stability is that a given perturbation is asymptotic to an almost periodic function in the sense of Bohr at infinity. Our results hold true with no assumptions on the smallness of given perturbations, and include the curve shortening flow problem as a special case.
2006, 14(1): 221-234
doi: 10.3934/dcds.2006.14.221

This paper is devoted to analyze a case of singularity formation in infinite time for a semilinear heat equation involving linear diffusion and superlinear convection. A feature to be noted is that blow-up happens not for the main unknown but for its derivative. The singularity builds up at the boundary. The formation of inner and outer regions is examined, as well as the matching between them. As a consequence, we obtain the precise exponential rates of blow-up in infinite time.

Select all articles

Export/Reference:

*+*[Abstract](393)

*+*[PDF](31.6KB)

**Abstract:**

Interaction between nonlinearity and diffusion is a fascinating subject. Many interesting phenomena are known to result from such interaction, for example, generation of interfaces and singularities, formation of spatial and temporal patterns, propagation of waves, symmetrization and symmetry-breaking of solutions, and so on. Nonlinear diffusive systems have undergone a thorough investigation aimed at mathematical understanding of the mechanisms behind the phenomena.

The analysis serving this aim uses various classical and newly developed techniques relying on results from the bifurcation theory, singular perturbation theory, variational method, and the theory of finite and infinite dimensional dynamical systems. The qualitative theory of parabolic and elliptic equations has been developing extensively in the last few decades, and a lot of important, interesting and beautiful results have been obtained concerning the dynamics of solutions and qualitative description of steady states.

For more information please click the “Full Text” above.

*+*[Abstract](589)

*+*[PDF](350.4KB)

**Abstract:**

We study the degenerate logistic model described by the equation $ u_t - $Δ$ u=au-b(x)u^p$ with standard boundary conditions, where $p>1$, $b$ vanishes on a nontrivial subset $\Omega_0$ of the underlying bounded domain $\Omega\subset R^N$ and $b$ is positive on $\Omega_+=\Omega\setminus \overline{\Omega}_0$. We consider the difficult case where $\partial\Omega_0\cap \partial \Omega$≠$\emptyset$ and $\partial\Omega_+\cap \partial \Omega$≠$\emptyset$, and examine the asymptotic behaviour of the solutions. By a detailed study of a singularly mixed boundary blow-up problem, we obtain some basic results on the dynamics of the model.

*+*[Abstract](695)

*+*[PDF](2728.1KB)

**Abstract:**

This paper is concerned with the dynamics of travelling spot solutions in two dimensions. Travelling spot solutions are constructed under the bifurcation structure with Jordan block type degeneracy. It is shown that if the velocity is very slow, such travelling spots possess reflection property. In order to do it, we derive the reduced ordinary differential equations describing the dynamics of interacting travelling spots in RD systems by using center manifold theory. This reduction enables us to prove that two very slowly travelling spots reflect before collision as if they were elastic particles.

*+*[Abstract](609)

*+*[PDF](230.5KB)

**Abstract:**

This paper examines the following question: Suppose that we have a reaction-diffusion equation or system such that some solutions which are homogeneous in space blow up in finite time. Is it possible to inhibit the occurrence of blow-up as a consequence of imposing Dirichlet boundary conditions, or other effects where diffusion plays a role? We give examples of equations and systems where the answer is affirmative.

*+*[Abstract](563)

*+*[PDF](303.9KB)

**Abstract:**

In this paper we study solutions to reaction-diffusion equations in the bistable case, defined on the whole space in dimension $N$. The existence of solutions with cylindric symmetry is already known. Here we prove the uniqueness of these cylindric solutions whose level sets are curved Lipschitz graphs. Using a centre manifold-like argument, we also give the precise asymptotics of these level sets at infinity. In dimension 2, we classify all solutions under weak conditions at infinity. Finally, we also provide an alternative proof of the existence of these solutions in dimension 2, based on a continuation argument.

*+*[Abstract](471)

*+*[PDF](331.9KB)

**Abstract:**

We study the existence of multiple positive stable solutions for

$ -\epsilon^2\Delta u(x) = u(x)^2(b(x)-u(x)) \ \mbox{in}\ \Omega, \quad$ $ \frac{\partial u}{\partial n}(x) = 0 \ \mbox{on}\ \partial\Omega.$

Here $\epsilon>0$ is a small parameter and $b(x)$ is a piecewise continuous function which changes sign. These type of equations appear in a population growth model of species with a saturation effect in biology.

*+*[Abstract](507)

*+*[PDF](301.6KB)

**Abstract:**

The blowup behaviors of solutions to a scalar-field equation with the Robin condition are discussed. For some range of the parameter, there exist at least two positive solutions to the equation. Here, the blowup rate of the large solution and the scaling properties are discussed.

*+*[Abstract](682)

*+*[PDF](261.6KB)

**Abstract:**

In this paper, we consider a Lotka-Volterra competition model with diffusion, and show that the global bifurcation structure of positive stationary solutions for the model is similar to that for a certain scalar reaction-diffusion equation. To do this, the comparison principle, the bifurcation theory, and the numerical verification are employed.

*+*[Abstract](542)

*+*[PDF](312.6KB)

**Abstract:**

We study a Ginzburg-Landau energy in a one-dimensional ring with nonuniform thickness, where the nonuniformity is expressed by a piecewise-constant function. That is a simplified model describing a supercurrent in the superconducting ring. Then the Ginzburg-Landau equation with a discontinuous coefficient subject to periodic boundary conditions is derived as the Euler-Lagrange equation of the energy functional. Since the unknown variable of the equation is complex-valued, we can define the phase of a solution if the solution has no zero. The purpose of this article is to establish the existence of nontrivial solutions with no zero and to reveal the configuration of the phase of the solutions as the coefficient converges to zero in a set of subintervals. More precisely we control the convergence of the coefficient with a small positive parameter $\varepsilon$ having various orders in the subintervals and prove the convergence of the solutions to those of a limiting equation as $\varepsilon\to0$ together with the convergence rate. As a consequence, for small $\varepsilon$ most of the phase variation takes place on the subintervals where the coefficient converges to zero with the highest order. Finally we show the stability of those solutions.

*+*[Abstract](811)

*+*[PDF](290.4KB)

**Abstract:**

We study the continuation of solutions of superlinear indefinite parabolic problems after the blow-up time. The nonlinearity is of the form $a(x)u^p$, where $p>1$ is subcritical and $a$ changes sign. Unlike the case $a>0$, the solutions will never blow up completely in the whole domain but only in a certain subdomain. In some cases we give a precise description of this subdomain. We also derive sufficient conditions for the blow-up of the associated energy.

*+*[Abstract](520)

*+*[PDF](254.2KB)

**Abstract:**

We consider positive solutions of the equation $- $ε$^2 $Δ $u + u $=$u^p$ in $\Omega$, where $\Omega \subseteq \R^n$, $p > 1$ and ε is a small positive parameter. Neumann boundary conditions are imposed in general. We prove existence of solutions which concentrate at curves or manifolds in $\overline{\Omega}$ when ε → 0.

*+*[Abstract](534)

*+*[PDF](289.5KB)

**Abstract:**

This paper is concerned with the long time behavior for the evolution of a curve governed by the curvature flow with constant driving force in two-dimensional space. Especially, the asymptotic stability of a traveling wave whose shape is a line is studied. We deal with moving curves represented by the entire graphs on the $x$-axis. By studying the Cauchy problem, the asymptotic stability of traveling waves with spatially decaying initial perturbations and the convergence rate are obtained. Moreover we establish the stability result where initial perturbations do not decay to zero but oscillate at infinity. In this case, we prove that one of the sufficient conditions for asymptotic stability is that a given perturbation is asymptotic to an almost periodic function in the sense of Bohr at infinity. Our results hold true with no assumptions on the smallness of given perturbations, and include the curve shortening flow problem as a special case.

*+*[Abstract](538)

*+*[PDF](270.7KB)

**Abstract:**

This paper is devoted to analyze a case of singularity formation in infinite time for a semilinear heat equation involving linear diffusion and superlinear convection. A feature to be noted is that blow-up happens not for the main unknown but for its derivative. The singularity builds up at the boundary. The formation of inner and outer regions is examined, as well as the matching between them. As a consequence, we obtain the precise exponential rates of blow-up in infinite time.

**
**

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