Degenerate Parabolic Systems Research Articles

Gradient Regularity for a Class of Widely Degenerate Parabolic Systems

Gradient Regularity for a Class of Widely Degenerate Parabolic Systems

Partial regularity for degenerate parabolic systems with general growth via caloric approximations

Partial regularity for degenerate parabolic systems with general growth via caloric approximations

Large-time asymptotics for degenerate cross-diffusion population models with volume filling

The large-time asymptotics of the solutions to a class of degenerate parabolic cross-diffusion systems is analyzed. The equations model the interaction of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. Compared to previous works, we allow for different diffusivities and degenerate nonlinearities. The proof is based on the relative entropy method, but in contrast to usual arguments, the relative entropy and entropy production are not directly related by a logarithmic Sobolev inequality. The key idea is to apply convex Sobolev inequalities to modified entropy densities including “iterated” degenerated functions.

Gradient Higher Integrability for Degenerate Parabolic Double-Phase Systems

We prove a local higher integrability result for the gradient of a weak solution to degenerate parabolic double-phase systems of p-Laplace type. This result comes with reverse Hölder type estimates. The proof is based on a careful phase analysis, estimates in the intrinsic geometries and stopping time arguments.

The principal eigenvalue for partially degenerate and periodic reaction-diffusion systems with time delay

In this paper, we first establish the theory of principal eigenvalues for a large class of partially degenerate, linear and periodic parabolic cooperative systems with time delay. Then we apply these theoretical results to study the global dynamics of a blacklegged tick Ixodes scapularis population model. To present a threshold-type result in terms of basic reproduction ratio R0 for such a model, we also extend the earlier theory of R0 to abstract functional differential equations with time-delayed internal transition.

The existence of continuous weak solutions with compact support for degenerate parabolic systems with exchange-type source

The presence of exchange-type source is an important feature of the radiation diffusion model coupled with material heat transfer. In this paper, we prove the existence of continuous weak solutions with compact support for the Cauchy problem of two classes of degenerate diffusion systems with exchange-type sources. One class is weakly coupled and another is special strongly coupled. For special strongly coupled degenerate systems, we further prove that the support of radiation is concurrent with that of material temperature, and the supports are extending as time goes on, which can be derived as a consequence of the exchange mechanism.

Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems

Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.

Gradient Hölder regularity for degenerate parabolic systems

We establish the local Hölder regularity of the spatial gradient of weak solutions to a class of general, degenerate parabolic systems. Our results can be regarded as a generalization of the fundamental work by DiBenedetto and Friedman (1985).

Blow-up behavior for a degenerate parabolic systems subject to Neumann boundary conditions

The degenerate parabolic systems dealt with in this paper are as follows: where r and s are two positive constants, spatial region is bounded, and is smooth. The conditions that make the solution blow up at a certain finite time are obtained. An upper bound of and an upper estimate of the blow-up rate of are also derived.

Partial regularity for degenerate parabolic systems with nonstandard growth and discontinuous coefficients

This article studies the partial Hölder continuity of weak solutions to certain degenerate parabolic systems whose model is the parabolic p(x,t)-Laplacian system,∂tu−div[μ(z)(1+|Du|2)p(z)−22Du]=0,p(z)≥2. Here, the exponential function p(z) satisfies a logarithmic continuity condition. We show that if μ(z) satisfies a certain VMO-type condition, then u is locally Hölder continuous except for a measure zero set.

Properties of generalized degenerate parabolic systems

Abstract In this article, we consider the parabolic system ( u i ) t = ∇ ⋅ ( m U m − 1 A ( ∇ u i , u i , x , t ) + ℬ ( u i , x , t ) ) , ( 1 ≤ i ≤ k ) {({u}^{i})}_{t}=\nabla \cdot (m{U}^{m-1}{\mathcal{A}}(\nabla {u}^{i},{u}^{i},x,t)+{\mathcal{ {\mathcal B} }}({u}^{i},x,t)),\hspace{1.0em}(1\le i\le k) in the range of exponents m > n − 2 n m\gt \frac{n-2}{n} where the diffusion coefficient U U depends on the components of the solution u = ( u 1 , … , u k ) {\bf{u}}=({u}^{1},\ldots ,{u}^{k}) . Under suitable structure conditions on the vector fields A {\mathcal{A}} and ℬ {\mathcal{ {\mathcal B} }} , we first showed the uniform L ∞ {L}^{\infty } boundedness of the function U U for t ≥ τ > 0 t\ge \tau \gt 0 . We also proved the law of L 1 {L}^{1} mass conservation and the local continuity of solution u {\bf{u}} . In the last result, all components of the solution u {\bf{u}} have the same modulus of continuity if the ratio between U U and u i {u}^{i} , ( 1 ≤ i ≤ k 1\le i\le k ), is uniformly bounded above and below.

Analysis of the null controllability of degenerate parabolic systems of Grushin type via the moments method

In this article, we compute the exact value of the minimal null control time for the Grushin equation controlled on a strip. This result is already known from a recent work by Beauchard et al. (Ann. Inst. Fourier 70(1), 2020); however, we propose a different approach based on the moments method. Our approach requires a careful spectral analysis of a truncated harmonic oscillator. We have also extended known results on biorthogonal families to real exponentials in the absence of a gap condition in order to obtain uniform estimates with respect to the asymptotic behavior of the associated counting function. As a by-product of our approach, we also characterize the minimal null control time for systems of coupled Grushin equations with a distributed control on a strip or with a boundary control.

Blow-up analysis of solutions for weakly coupled degenerate parabolic systems with nonlinear boundary conditions

This paper is devoted to the study of the blow-up solutions of the following weakly coupled degenerate parabolic systems with nonlinear boundary conditions: ut=div|∇u|p∇u+f(x,u,v,t),vt=div|∇v|q∇v+g(x,u,v,t),(x,t)∈D×(0,T∗),∂u∂n=b(u),∂v∂n=d(v),(x,t)∈∂D×(0,T∗),u(x,0)=u0(x),v(x,0)=v0(x),x∈D¯.Here p>0,q>0, D is a bounded spatial region in Rn(n≥2), and the boundary ∂D is smooth. We mainly combine the maximum principles of the weakly coupled parabolic systems with the first-order differential inequality technique to discuss the blow-up phenomenon of the above problem. Sufficient conditions for the blow-up of the nonnegative solution (u,v) of this problem are given. In addition, for the nonnegative blow-up solution (u,v), we also obtain an upper bound on the blow-up time and an upper estimate of the blow-up rate.

The Cauchy problem for a nonlinear degenerate parabolic system in non-divergence form

We deal with degenerate quasilinear parabolic systems in the non-divergence form under positive initial conditions. An asymptotic behavior of self-similar solutions in the case of slow diffusion is established. Depending on values of the numerical parameters and the initial value, the existence of the global solutions of the Cauchy problem is proved. In addition, the asymptotic representation of the solution is obtained.

Liouville type theorems for degenerate parabolic systems with advection terms

We study a parabolic equation of the form $$\begin{aligned} u_t - \Delta _\lambda u+a\cdot \nabla _\lambda u=u^p \hbox { in}\,\,{\mathbb {R}}^{{N}}\times {\mathbb {R}}, \end{aligned}$$ and a parabolic system $$\left\{ {\begin{array}{*{20}l} {u_{t} - \Delta _{\lambda } u + a \cdot \nabla _{\lambda } u = v^{p} } \hfill \\ {u_{t} - \Delta _{\lambda } v + a \cdot \nabla _{\lambda } v = u^{q} } \hfill \\ \end{array} } \right.{\text{in}}\;\mathbb{R}^{N} \times \mathbb{R},$$ where p is a real number, a(x) is a smooth vector field, $$\Delta _\lambda$$ is a strongly degenerate operator given by $$\begin{aligned} \Delta _\lambda =\sum _{i=1}^N\partial _{x_i}\left( \lambda _i^2\partial _{x_i}\right) \end{aligned}$$ and $$\nabla _\lambda$$ is the gradient operator associated to $$\Delta _\lambda$$ . Under some general condition of $$\lambda _i$$ introduced in Kogoj and Lanconelli [Nonlinear Anal 75: 4637–4649, 2012], we establish a Liouville type theorem for positive supersolutions to the problems above. In particular, we compute explicitly the critical exponent depending on both the homogeneous dimension of $${\mathbb {R^N}}$$ associated to $$\Delta _\lambda$$ and the behavior of the advection term a at infinity. This critical exponent is sharp in the case of Laplace operator without advection term according to [15].

Degenerate Parabolic Systems of the Diffusion Type with Inertia

For a class of degenerate parabolic Kolmogorov-type systems with degeneration and arbitrarily (but finitely) many groups of degeneration variables and time-dependent coefficients of the parabolic part, we study the Cauchy problem, construct its fundamental matrix of solutions, and establish the estimates for its derivatives.

Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component

This article presents an inverse source problem for a cascade system of \begin{document}$ n $\end{document} coupled degenerate parabolic equations. In particular, we prove stability and uniqueness results for the inverse problem of determining the source terms by observations in an arbitrary subdomain over a time interval of only one component and data of the \begin{document}$ n $\end{document} components at a fixed positive time \begin{document}$ T' $\end{document} over the whole spatial domain. The proof is based on the application of a Carleman estimate with a single observation acting on a subdomain.

Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix

In this paper we study the null controllability of some non diagonalizable degenerate parabolic systems of PDEs, we assume that the diffusion, coupling and controls matrices are constant and we characterize the null controllability by an algebraic condition so called Kalman's rank condition.

Local continuity and asymptotic behaviour of degenerate parabolic systems

We study the local Hölder continuity and the asymptotic behaviour of solution, u=(u1,…,uk), of the degenerate system uti=∇⋅mUm−1∇uifor m>1 and i=1,…,kwhich describes the population densities of k-species whose diffusions are determined by their total population density U=u1+⋯+uk. For the local Hölder continuity, we adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and DiBenedetto. Under some regularity conditions, we also prove that the population density of ith species with the population Mi converges in Cs∞ to the function MiMBM(x,t) as t→∞ where BM is the Barenblatt profile of the standard porous medium equation with L1 mass M=M1+⋯+Mk. As a consequence of asymptotic behaviour, it is shown that each density becomes a concave function after a finite time.

On periodic subsolutions to steady second-order systems and applications

Bostan and Namah (Remarks on bounded solutions of steady Hamilton–Jacobi equations, C. R. Acad. Sci. Paris, Ser. I 347(15–16) (2009) 873–878) proved that constant functions are the only bounded solutions to H(Du)=H(0) when H is superlinear and strictly convex. In this short note, we present a proof other than that of Bostan and Namah for equations that can be easily applied to some types of possibly degenerate parabolic systems. Our proof applies for periodic subsolutions instead of bounded solutions like that of Bostan and Namah; however, we need periodic subsolutions, which is quite restrictive. We do not consider Hopf–Lax's formula in our proof, so we can relax some restrictions on H. We also present an application to the large-time behavior of solutions to degenerate parabolic systems.