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Discrete & Continuous Dynamical Systems - A

2010 , Volume 26 , Issue 2

Special Issue on Nonlinear parabolic problems and applications

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Patrick Guidotti and  Gieri Simonett
2010, 26(2): i-iii doi: 10.3934/dcds.2010.26.2i +[Abstract](26) +[PDF](52.6KB)
Nonlinear parabolic problems and applications. Nonlinear parabolic problems have long attracted the attention of the mathematical community, not least because mathematical models in several applied sciences often lead to nonlinear reaction-diffusion equations or systems. This volume is dedicated to both theoretical advances in the understanding of this broad class of problems and to the analysis of specific models of practical relevance. The 17 papers which make up the special issue can broadly be divided into two categories. One comprises papers which are mainly concerned with existence or regularity theory. Qualitative properties of solutions are the central focus of the other.
   The first paper, co-authored by Pablo Álvarez-Caudevilla and Julián López-Gómez, deals with a class of sublinear parabolic systems. A bifurcation analysis is performed which fully describes the asymptotic behavior of the system. The paper represents an important step towards a complete understanding of cooperative reaction-diffusion systems.

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The dynamics of a class of cooperative systems
Pablo Álvarez-Caudevilla and  Julián López-Gómez
2010, 26(2): 397-415 doi: 10.3934/dcds.2010.26.397 +[Abstract](41) +[PDF](254.0KB)
This paper characterizes the dynamics of the positive solutions of a class of sublinear parabolic cooperative systems whose analysis is imperative for ascertaining the dynamics of wider classes of sublinear cooperative and superlinear indefinite Reaction-Diffusion systems, like those introduced by López-Gómez and Molina-Meyer [15].
Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system
Tomasz Cieślak and  Philippe Laurençot
2010, 26(2): 417-430 doi: 10.3934/dcds.2010.26.417 +[Abstract](31) +[PDF](224.6KB)
It is known that classical solutions to the one-dimensional quasilinear Smoluchowski-Poisson system with nonlinear diffusion $a(u)=(1+u)^{-p}$ may blow up in finite time if $p>1$ and exist globally if $p<1$. The case $p=1$ thus appears to be critical but it turns out that all solutions are global also in that case. Two classes of diffusion coefficients are actually identified in this paper, one for which all solutions to the corresponding quasilinear Smoluchowski-Poisson system are global and the other one leading to finite time blowup for sufficiently concentrated initial data. The cornerstone of the proof are an alternative formulation of the Smoluchowski-Poisson system which relies on a novel change of variables and a virial identity.
The surface diffusion flow on rough phase spaces
Joachim Escher and  Piotr B. Mucha
2010, 26(2): 431-453 doi: 10.3934/dcds.2010.26.431 +[Abstract](29) +[PDF](323.8KB)
The surface diffusion flow is the gradient flow of the surface functional of compact hypersurfaces with respect to the inner product of $H^{-1}$ and leads to a nonlinear evolution equation of fourth order. This is an intrinsically difficult problem, due to the lack of an maximum principle and it is known that this flow may drive smoothly embedded uniformly convex initial surfaces in finite time into non-convex surfaces before developing a singularity [15, 16]. On the other hand it also known that singularities may occur in finite time for solutions emerging from non-convex initial data, cf. [10].
   Combining tools from harmonic analysis, such as Besov spaces, multiplier results with abstract results from the theory of maximal regularity we present an analytic framework in which we can investigate weak solutions to the original evolution equation. This approach allows us to prove well-posedness on a large (Besov) space of initial data which is in general larger than $C^2$ (and which is in the distributional sense almost optimal). Our second main result shows that the set of all compact embedded equilibria, i.e. the set of all spheres, is an invariant manifold in this phase space which attracts all solutions which are close enough (which respect to the norm of the phase space) to this manifold. As a consequence we are able to construct non-convex initial data which generate global solutions, converging finally to a sphere.
Low Mach number asymptotics for reacting compressible fluid flows
Eduard Feireisl and  Hana Petzeltová
2010, 26(2): 455-480 doi: 10.3934/dcds.2010.26.455 +[Abstract](33) +[PDF](337.4KB)
We study the low Mach number limit for the full Navier-Stokes-Fourier system describing the dynamics of chemically reacting fluids. The so-called reactive Boussinesq system is identified as the asymptotic limit.
Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations
Simona Fornaro and  Ugo Gianazza
2010, 26(2): 481-492 doi: 10.3934/dcds.2010.26.481 +[Abstract](28) +[PDF](180.5KB)
In the present paper we study the local behavior of non-negative weak solutions of a wide class of doubly non linear degenerate parabolic equations. We show, in particular, some lower pointwise estimates of such solutions in terms of suitable sub-potentials (dictated by the structure of the equation) and an alternative form of the Harnack inequality.
Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary
Yoshikazu Giga , Przemysław Górka and  Piotr Rybka
2010, 26(2): 493-519 doi: 10.3934/dcds.2010.26.493 +[Abstract](37) +[PDF](354.0KB)
We consider the weighted mean curvature flow in the plane with a driving term. For certain anisotropy functions this evolution problem degenerates to a first order Hamilton-Jacobi equation with a free boundary. The resulting problem may be written as a Hamilton-Jacobi equation with a spatially non-local and discontinuous Hamiltonian. We prove existence and uniqueness of solutions. On the way we show a comparison principle and a stability theorem for viscosity solutions.
Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density
Shoshana Kamin , Guillermo Reyes and  Juan Luis Vázquez
2010, 26(2): 521-549 doi: 10.3934/dcds.2010.26.521 +[Abstract](69) +[PDF](379.2KB)
We study the long-time behavior of nonnegative solutions to the Cauchy problem

$ \rho(x)\, \partial_t u= \Delta u^m $ in $Q$:$=\mathbb{R}^N\times\mathbb{R}_+$
$u(x, 0)=u_0 $ in $\mathbb{R}^N$

in dimensions $ N\ge 3$. We assume that $ m> 1 $ and $ \rho(x) $ is positive and bounded with $ \rho(x)\le C|x|^{-\gamma} $ as $ |x|\to\infty$ with $\gamma>2$. The initial data $u_0$ are nonnegative and have finite energy, i.e., $ \int \rho(x)u_0 dx< \infty$.
   We show that in this case nontrivial solutions to the problem have a long-time universal behavior in separate variables of the form

$u(x,t)$~$ t^{-1/(m-1)}W(x),$

where $V=W^m$ is the unique bounded, positive solution of the sublinear elliptic equation $-\Delta V=c\,\rho(x)V^{1/m}$, in $\mathbb{R}^N$ vanishing as $|x|\to\infty$; $c=1/(m-1)$. Such a behavior of $u$ is typical of Dirichlet problems on bounded domains with zero boundary data. It strongly departs from the behavior in the case of slowly decaying densities, $\rho(x)$~$ |x|^{-\gamma}$ as $|x|\to\infty$ with 0 ≤ $ \gamma<2$, previously studied by the authors.
   If $\rho(x)$ has an intermediate decay, $\rho$~$ |x|^{-\gamma}$ as $|x|\to\infty$ with $2<\gamma<\gamma_2$:$=N-(N-2)/m$, solutions still enjoy the finite propagation property (as in the case of lower $\gamma$). In this range a more precise description may be given at the diffusive scale in terms of source-type solutions $U(x,t)$ of the related singular equation $|x|^{-\gamma}u_t= \Delta u^m$. Thus in this range we have two different space-time scales in which the behavior of solutions is non-trivial. The corresponding results complement each other and agree in the intermediate region where both apply, thus providing an example of matched asymptotics.

Random dispersal vs. non-local dispersal
Chiu-Yen Kao , Yuan Lou and  Wenxian Shen
2010, 26(2): 551-596 doi: 10.3934/dcds.2010.26.551 +[Abstract](51) +[PDF](500.9KB)
Random dispersal is essentially a local behavior which describes the movement of organisms between adjacent spatial locations. However, the movements and interactions of some organisms can occur between non-adjacent spatial locations. To address the question about which dispersal strategy can convey some competitive advantage, we consider a mathematical model consisting of one reaction-diffusion equation and one integro-differential equation, in which two competing species have the same population dynamics but different dispersal strategies: the movement of one species is purely by random walk while the other species adopts a non-local dispersal strategy. For spatially periodic and heterogeneous environments we show that (i) for fixed random dispersal rate, if the nonlocal dispersal distance is sufficiently small, then the non-local disperser can invade the random disperser but not vice versa; (ii) for fixed nonlocal dispersal distance, if the random dispersal rate becomes sufficiently small, then the random disperser can invade the nonlocal disperser but not vice versa. These results suggest that for spatially periodic heterogeneous environments, the competitive advantage may belong to the species with much lower effective rate of dispersal. This is in agreement with previous results for the evolution of random dispersal [9, 13] that the slower disperser has an advantage. Nevertheless, if random dispersal strategy with either zero Dirichlet or zero Neumann boundary condition is compared with non-local dispersal strategy with hostile surroundings, the species with much lower effective rate of dispersal may not have the competitive advantage. Numerical results will be presented to shed light on the global dynamics of the system for general values of non-local interaction distance and also to point to future research directions.
Higher integrability for gradients of solutions to degenerate parabolic systems
Dung Le
2010, 26(2): 597-608 doi: 10.3934/dcds.2010.26.597 +[Abstract](33) +[PDF](170.8KB)
Using the method of heat approximation, we will establish higher integrability for the gradients of bounded weak solutions to certain strongly coupled degenerate parabolic systems.
An ill-posed problem for the Navier-Stokes equations for compressible flows
Misha Perepelitsa
2010, 26(2): 609-623 doi: 10.3934/dcds.2010.26.609 +[Abstract](36) +[PDF](246.7KB)
We consider the Navier-Stokes equations for the motion of a compressible, viscous, isentropic fluid in a half-space H2. We prove that under the no-slip boundary conditions, the initial-boundary value problem is ill-posed in the space ($\rho$($t,\cdot$)$,\grad_x\u$($t,\cdot$))$\in$($L^\infty_x$(H2)$\times L^\infty_x$(H2))$.$
Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory
Jan Prüss , Vicente Vergara and  Rico Zacher
2010, 26(2): 625-647 doi: 10.3934/dcds.2010.26.625 +[Abstract](38) +[PDF](334.3KB)
In this paper we study a temperature dependent phase field model with memory. The case where both the equation for the temperature and that for the order parameter is of fractional time order is covered. Under physically reasonable conditions on the nonlinearities we prove global well-posedness in the $L_p$ setting and show that each solution converges to a steady state as time goes to infinity.
Wellposedness of the tornado-hurricane equations
Jürgen Saal
2010, 26(2): 649-664 doi: 10.3934/dcds.2010.26.649 +[Abstract](34) +[PDF](228.1KB)
We prove local-in-time existence of a unique mild solution for the tornado-hurricane equations in a Hilbert space setting. The wellposedness is shown simultaneously in a halfspace, a layer, and a cylinder and for various types of boundary conditions which admit discontinuities at the edges of the cylinder. By an approach based on symmetric forms we first prove maximal regularity for a linearized system. An application of the contraction mapping principle then yields the existence of a unique local-in-time mild solution.
Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source
Kaouther Ammar and  Philippe Souplet
2010, 26(2): 665-689 doi: 10.3934/dcds.2010.26.665 +[Abstract](32) +[PDF](319.7KB)
We prove universal bounds for nonnegative weak solutions of the porous medium equation with source $u_t-\Delta u^m=u^p$ where $1 < m < p$. These bounds imply initial and final blow-up rate estimates, as well as a~priori estimates or decay rates for global solutions. We consider both radial and nonradial solutions, and in the radial case we cover all Sobolev-subcritical values of $p/m$, which is the best possible range. Our bounds are proved as a consequence of Liouville-type theorems for entire solutions and doubling and rescaling arguments. In this connection, we use known Liouville-type theorems for radial solutions, along with some new Liouville-type theorems that are here established for nonradial solutions in RN and for solutions on a half-line.
Age-dependent equations with non-linear diffusion
Christoph Walker
2010, 26(2): 691-712 doi: 10.3934/dcds.2010.26.691 +[Abstract](31) +[PDF](307.2KB)
We consider the well-posedness of models involving age structure and non-linear diffusion. Such problems arise in the study of population dynamics. It is shown how diffusion and age boundary conditions can be treated that depend non-linearly and possibly non-locally on the density itself. The abstract approach is applied to concrete examples.
Boundary dynamics of a two-dimensional diffusive free boundary problem
Micah Webster and  Patrick Guidotti
2010, 26(2): 713-736 doi: 10.3934/dcds.2010.26.713 +[Abstract](37) +[PDF](265.2KB)
Numerous models of industrial processes such as diffusion in glassy polymers or solidification phenomena, lead to general one-phase free boundary value problems with phase onset. In this paper we develop a framework viable to prove global existence and stability of planar solutions to one such multi-dimensional model whose application is in controlled-release pharmaceuticals. We utilize a boundary integral reformulation to allow for the use of maximal regularity. To this effect, we view the operators as pseudo-differential and exploit knowledge of the relevant symbols. Within this framework, we give a local existence and continuous dependence result necessary to prove planar solutions are locally exponentially stable with respect to two-dimensional perturbations.
Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth
Junde Wu and  Shangbin Cui
2010, 26(2): 737-765 doi: 10.3934/dcds.2010.26.737 +[Abstract](36) +[PDF](372.7KB)
In this paper we first study local phase diagram of an abstract parabolic differential equation in a Banach space such that the equation possesses an invariance structure under a local Lie group action. Next we use this abstract result to study a free boundary problem modeling the growth of non-necrotic tumors in the presence of inhibitors. This problem contains two reaction-diffusion equations describing the diffusion of the nutrient and the inhibitor, respectively, and an elliptic equation describing the distribution of the internal pressure. There is also an equation for the surface tension to govern the movement of the free boundary. By first performing some reduction processes to write this free boundary problem into a parabolic differential equation in a Banach space, and next using a new center manifold theorem established recently by Cui [8], and the abstract result mentioned above, we prove that under suitable conditions the radial stationary solution is locally asymptotically stable under small non-radial perturbations, and when these conditions are not satisfied then such a stationary solution is unstable. In the second case we also give a description of local phase diagram of the equation in a neighborhood of the radial stationary solution and construct its stable and unstable manifolds. In particular, we prove that in the unstable case, if the transient solution exists globally and is contained in a neighborhood of the radial stationary solution, then the transient solution will finally converge to a nearby radial stationary solution uniquely determined by the initial data.
Gradient blowup rate for a semilinear parabolic equation
Zhengce Zhang and  Bei Hu
2010, 26(2): 767-779 doi: 10.3934/dcds.2010.26.767 +[Abstract](40) +[PDF](195.0KB)
We present a one-dimensional semilinear parabolic equation $u_t=$u xx$ +x^m |u_x|^p, p> 0, m\geq 0$, for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We show that the spatial derivative of solutions is globally bounded in the case $p\leq m+2$ while blowup occurs at the boundary when $p>m+2$. Blowup rate is also found for some range of $p$.

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