We consider weak solutions u:ΩT→RN to parabolic systems of the typeut−divA(x,t,Du)=finΩT=Ω×(0,T), where Ω is a bounded open subset of Rn for n≥2, T>0 and the datum f belongs to a suitable Orlicz ...space. The main novelty here is that the partial map ξ↦A(x,t,ξ) satisfies standard p-growth and ellipticity conditions for p>1 only outside the unit ball {|ξ|<1}. For p>2nn+2 we establish that any weak solutionu∈C0((0,T);L2(Ω,RN))∩Lp(0,T;W1,p(Ω,RN)) admits a locally bounded spatial gradient Du. Moreover, assuming that u is essentially bounded, we recover the same result in the case 1<p≤2nn+2 and f=0. Finally, we also prove the uniqueness of weak solutions to a Cauchy-Dirichlet problem associated with the parabolic system above. We emphasize that our results include both the degenerate case p≥2 and the singular case 1<p<2.
Models of tissue growth are now well established, in particular, in relation to their applications to cancer. They describe the dynamics of cells subject to motion resulting from a pressure gradient ...generated by the death and birth of cells, itself controlled primarily by pressure through contact inhibition. In the compressible regime, pressure results from the cell densities and when two different populations of cells are considered, a specific difficulty arises; the equation for each cell density carries a hyperbolic character, and the equation for the total cell density has a degenerate parabolic property. For that reason, few a priori estimates are available and discontinuities may occur. Therefore the existence of solutions is a difficult problem. Here, we establish the existence of weak solutions to the model with two cell populations which react similarly to the pressure in terms of their motion but undergo different growth/death rates. In opposition to the method used in the recent paper of Carrillo et al., our strategy is to ignore compactness of the cell densities and to prove strong compactness of the pressure gradient. For that, we propose a new version of Aronson-Bénilan estimate, working in L
2
rather than
L
∞
.
We improve known results in three directions; we obtain new estimates, we treat dimensions higher than 1 and we deal with singularities resulting from vacuum.
In this paper we study this chemotaxis-system{ut=Δu−χ∇⋅(u∇v)+g(u),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, defined in a convex smooth and bounded domain Ω of Rn, with n≥1, and endowed with homogeneous Neumann ...boundary conditions, being χ>0 and g a sort of logistic function obeying growth technical restrictions. Specifically, once an appropriate definition of very weak solution is given, our main result deals with the global existence of such solutions for any nonnegative initial data (u0,v0)∈C0(Ω¯)×C2(Ω¯), and under zero-flux boundary condition on v0.
We consider vector-valued solutions to a linear transmission problem, and we prove that Lipschitz-regularity on one phase is transmitted to the next phase. More exactly, given a solution u:B1⊂Rn→Rm ...to the elliptic system div((A+(B−A)χD)∇u)=0inB1,where A and B are Dini continuous, uniformly elliptic matrices, we prove that if ∇u∈L∞(D) then u is Lipschitz in B1/2. A similar result is also derived for the parabolic counterpart of this problem.
In this article, the fuzzy dynamic event-triggered tracking control problem of semilinear parabolic systems (SLPSs) with time-varying delay is investigated. Firstly, T-S fuzzy partial differential ...equation (PDE) models are introduced to describe the SLPSs. Secondly, a less conservative and more general fuzzy dynamic event-triggered strategy (DETS) is proposed to reduce communication consumption and avoid unnecessary continuous signal monitoring. Since the dynamic threshold is closely related to the currently sampled signal and the latest successfully transmitted signal, it can be promptly dynamically adjusted. In addition, on the basis of a reasonable assumption, a novel linear matrix inequality (LMI) relax technique is introduced to deal with the mismatched premise variables between the fuzzy systems and the fuzzy controller. By constructing the appropriate Lyapunov-Krasovskii candidate functional (LKCF), the criteria that the SLPSs can asymptotically track the target systems is derived, and the desired dynamic event-triggered control (DETC) gains can be obtained by solving a set of LMIs. The DETS reduces effectively communication resource consumption. Finally, the control problem of temperature distribution of catalytic rod in practical engineering application is given to verify the effectiveness and superiority of the proposed control scheme.
We prove estimates of Calderón–Zygmund type for evolutionary p-Laplacian systems in the setting of Lorentz spaces. We suppose the coefficients of the system to satisfy only a VMO condition with ...respect to the space variable. Our results hold true, mutatis mutandis, also for stationary p-Laplacian systems.
We investigate degenerate cross-diffusion equations, with a rank-deficient diffusion-matrix, modelling multispecies population dynamics driven by partial pressure gradients. These equations have ...recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic-parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in
for
In this paper we study the quantitative homogenization of second-order parabolic systems with locally periodic (in both space and time) coefficients. The O(ε) scale-invariant error estimate in ...L2(0,T;L2dd−1(Ω)) is established in C1,1 cylinders under minimum smoothness conditions on the coefficients. This process relies on critical estimates of smoothing operators. We also develop a new construction of flux correctors in the parabolic manner and a sharp estimate for temporal boundary layers.
We consider quantitative estimates in the homogenization of second-order parabolic systems with periodic coefficients that oscillate on multiple spatial and temporal ...scales,∂t−div(A(x,t,x/ε1,…,x/εn,t/ε1′,…,t/εm′)∇), where εℓ=εαℓ,εk′=εβk,ℓ=1,...,n,k=1,...,m, with 0<α1<...<αn<∞ and 0<β1<...<βm<∞. The convergence rate in the homogenization is derived in the L2 space, and the large-scale interior and boundary Lipschitz estimates are also established. In the case n=m=1, such issues have been addressed by Geng and Shen (2020) 12 based on an interesting scale reduction technique developed therein. Our investigation relies on a quantitative reiterated homogenization theory.
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