# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

April 2010 , Volume 26 , Issue 2

Special Issue on Nonlinear parabolic problems and applications

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2010, 26(2): i-iii doi: 10.3934/dcds.2010.26.2i +[Abstract](1227) +[PDF](52.6KB)
Abstract:
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.

2010, 26(2): 397-415 doi: 10.3934/dcds.2010.26.397 +[Abstract](1554) +[PDF](254.0KB)
Abstract:
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].
2010, 26(2): 417-430 doi: 10.3934/dcds.2010.26.417 +[Abstract](1424) +[PDF](224.6KB)
Abstract:
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.
2010, 26(2): 431-453 doi: 10.3934/dcds.2010.26.431 +[Abstract](1568) +[PDF](323.8KB)
Abstract:
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.
2010, 26(2): 455-480 doi: 10.3934/dcds.2010.26.455 +[Abstract](1527) +[PDF](337.4KB)
Abstract:
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.
2010, 26(2): 481-492 doi: 10.3934/dcds.2010.26.481 +[Abstract](1587) +[PDF](180.5KB)
Abstract:
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.
2010, 26(2): 493-519 doi: 10.3934/dcds.2010.26.493 +[Abstract](1704) +[PDF](354.0KB)
Abstract:
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.
2010, 26(2): 521-549 doi: 10.3934/dcds.2010.26.521 +[Abstract](1334) +[PDF](379.2KB)
Abstract:
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.

2010, 26(2): 551-596 doi: 10.3934/dcds.2010.26.551 +[Abstract](1899) +[PDF](500.9KB)
Abstract:
2010, 26(2): 597-608 doi: 10.3934/dcds.2010.26.597 +[Abstract](1234) +[PDF](170.8KB)
Abstract:
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.
2010, 26(2): 609-623 doi: 10.3934/dcds.2010.26.609 +[Abstract](1547) +[PDF](246.7KB)
Abstract:
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))$.$
2010, 26(2): 625-647 doi: 10.3934/dcds.2010.26.625 +[Abstract](1643) +[PDF](334.3KB)
Abstract:
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.
2010, 26(2): 649-664 doi: 10.3934/dcds.2010.26.649 +[Abstract](1370) +[PDF](228.1KB)
Abstract:
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.
2010, 26(2): 665-689 doi: 10.3934/dcds.2010.26.665 +[Abstract](1669) +[PDF](319.7KB)
Abstract:
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.
2010, 26(2): 691-712 doi: 10.3934/dcds.2010.26.691 +[Abstract](1351) +[PDF](307.2KB)
Abstract:
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.
2010, 26(2): 713-736 doi: 10.3934/dcds.2010.26.713 +[Abstract](1768) +[PDF](265.2KB)
Abstract:
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.
2010, 26(2): 737-765 doi: 10.3934/dcds.2010.26.737 +[Abstract](2279) +[PDF](372.7KB)
Abstract:
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.
2010, 26(2): 767-779 doi: 10.3934/dcds.2010.26.767 +[Abstract](1737) +[PDF](195.0KB)
Abstract:
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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