Karthik Adimurthi, Vivek Tewary
We obtain local boundedness of weak solutions of double phase quasilinear parabolic equations of the form \[u_t-\text{div} \left(|\nabla u|^{p-2}\nabla u+a(x,t)|\nabla u|^{q-2}\nabla u\right)=0,\] where, we have imposed the restrictions $\frac{2N}{N+2}<p<\infty$, $0\leq a(x,t)\leq M$ is measurable and $q < p\frac{N+1}{N-1}$.
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This article sets forth results on the existence, positivity and boundedness of solutions for quasilinear elliptic systems involving p-Laplacian and q-Laplacian operators. The approach combines Schaefer's fixed point, comparison principle as well as Moser's iteration procedure.
We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi's counterexample. Here we assume a condition on the support of off-diagonal coefficients that "keeps away" the counterexample and allows us to prove local boundedness of weak solutions.
Uniform $L^{\infty}$-boundedness for solutions of anisotropic quasilinear systems
本文研究了各向异性拟线性系统解的统一 $L^{\infty}$-有界性，通过非自主系统和临界增长非线性项的分析，得到解的局部有界估计。
We derive local boundedness estimates for weak solutions of a large class of second order quasilinear equations. The structural assumptions imposed on an equation in the class allow vanishing of the quadratic form associated with its principal part and require no smoothness of its coefficients. The class includes second order linear elliptic equations as studied by D. Gilbarg and N. S. Trudinger [1998] and second order subelliptic linear equations as studied by E. Sawyer and R. L. Wheeden [2006 and 2010]. Our results also extend ones obtained by J. Serrin [1964] concerning local boundedness of…
In this paper we consider a one-dimensional fully parabolic quasilinear Keller-Segel system with critical nonlinear diffusion. We show uniform-in-time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well-known Lyapunov functional and a recently introduced new Lyapunov-like functional in [3].
We consider the two-dimensional quasilinear wave equations with quadratic nonlinearities. We introduce a new class of null forms and prove uniform boundedness of the highest order norm of the solution for all time. This class of null forms include several prototypical strong null conditions as special cases. To handle the critical decay near the light cone we inflate the nonlinearity through a new normal form type transformation which is based on a deep cancelation between the tangential and normal derivatives with respect to the light cone. Our proof does not employ the Lorentz boost and can …
We provide sufficient and almost optimal conditions for global existence of classical solutions in parabolic Hölder spaces to quasilinear one-dimensional parabolic problems with dynamical boundary conditions.
We introduce a notion of quasilinear parabolic equations over metric measure spaces. Under sharp structural conditions, we prove that local weak solutions are locally bounded and satisfy the parabolic Harnack inequality. Applications include the parabolic maximum principle and pointwise estimates for weak solutions.
Finite time blow-up is shown to occur for solutions to a one-dimensional quasilinear parabolic-parabolic chemotaxis system as soon as the mean value of the initial condition exceeds some threshold value. The proof combines a novel identity of virial type with the boundedness from below of the Liapunov functional associated to the system, the latter being peculiar to the one-dimensional setting.
Nonlocal Quasilinear Parabolic Equations in Heisenberg Group: Local Boundedness with an Optimal Tail
本文在最优尾项正则性假设下证明了海森堡群上拟线性抛物方程解的局部有界性，并建立了分数阶 Sobolev 空间的插值不等式与延拓定理。
We establish a spatial gradient maximum principle for classical solutions to the initial and Neumann boundary value problem of some quasilinear parabolic equations on smooth convex domains.
In this paper, we study the boundedness of weak solutions to quasilinear parabolic equations of the form \[u_t - \text{div} \mathcal{A}(x,t,\nabla u) = 0, \] where the nonlinearity $\mathcal{A}(x,t,\nabla u)$ is modelled after the well studied $p$-Laplace operator. The question of boundedness has received lot of attention over the past several decades with the existing literature showing that weak solutions in either $\frac{2N}{N+2}<p<2$, $p=2$ or $2<p$ are bounded. The proof is essentially split into three cases mainly because the estimates that have been obtained in the past always included …
We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear dynamic boundary conditions.
Bounded smooth solutions of the Dirichlet and Neumann problems for a wide variety of quasilinear parabolic equations, including graphical anisotropic mean curvature flows, have gradient bounded in terms of oscillation and elapsed time.
Regimes with a singular peaking for a wide class of quasilinear second order parabolic equations are studied. On the basis of energy methods, precise estimates of a final profile of a weak solution in a neighborhood of the peaking time are established depending on the rate of increase of a global energy of this solution. Key words: quasilinear parabolic equations, peaking regimes, energy solutions.
Based on a comparison principle, we derive an exponential rate of convergence for solutions to the initial-boundary value problem for a class of quasilinear parabolic equations in one space dimension. We then apply the result to some models in population dynamics and image processing.
In this paper, we provide a direct approach to the existence and uniqueness of strong (in the probabilistic sense) and weak (in the PDE sense) solutions to quasilinear stochastic partial differential equations, which are neither monotone nor locally monotone.
Unboundedness of solutions is shown to occur in a one-dimensional quasilinear parabolicparabolic chemotaxis system for any initial mass. Our result is also independent of the relation between the speeds of the diffusion of cells and chemoattractant. The proof is achieved by contradiction. It uses a virial type inequality together with a boundedness from below of the Liapunov functional associated to the system. Moreover, the same method is applied also in higher dimensions and an unboundedness result is shown in the case of critical quasilinear fully parabolic Keller-Segel system for mass larg…
Nonautonomous maximal parabolic regularity for nonsmooth quasilinear parabolic systems
本文针对非光滑域上带混合边界条件的非自治拟线性抛物系统，建立了$L^p$-极大正则性，通过外推技术获得一致估计，仅需温和边界正则性及有界可测复系数，并允许时间非局部非线性，涵盖稳态不满足椭圆性的情形。
Asymptotic solutions of a quasilinear parabolic equation with a small parameter at the higher derivative are constructed near large-gradient and Lagrange singularities of A-type, which represent interest for studying processes of shock waves formation in physical media with a small nonzero viscosity.
We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.
Motivated by the study of conditional stability of traveling waves, we give an elementary $H^2$ center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and "damping-type" $H^s$ energy estimates of a type familiar from the study of hyperbolic--parabolic and relaxation systems. An important feature of these methods is that they generalize to situations such as the hyperbolic--parabolic or relaxation case for which parabolic-type smoothing estimates are unavailable. A…
We present in this note a local in time well-posedness result for the singular $2$-dimensional quasilinear generalized parabolic Anderson model equation $$ \partial_t u - a(u)Δu = g(u)ξ$$ The key idea of our approach is a simple transformation of the equation which allows to treat the problem as a semilinear problem. The analysis is done within the elementary setting of paracontrolled calculus.
We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.
We consider the quasilinear parabolic-parabolic Keller-Segel system $$ u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \qquad x\inΩ, \ t>0, v_t=Δv -v + u, x\inΩ, \ t>0, $$ under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset\R^n$ with $n\ge 2$. It is proved that if $\frac{S(u)}{D(u)}\le cu^α$ with $α<\frac{2}{n}$ and some constant $c>0$ for all $u>1$ and some further technical conditions are fulfilled, then the classical solutions to the above system are uniformly-in-time bounded. This boundedness result is optimal according to a recent result by th…
We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global existence of solutions and we extend the generalized principle of linearized stability to settings with initial values in critical spaces. These general results are applied to the surface diffusion flow in various settings.
This paper is concerned with a quaslinear parabolic equation including a nonlinear nonlocal initial condition. The problem arises as equilibrium equation in population dynamics with nonlinear diffusion. We make use of global bifurcation theory to prove existence of an unbounded continuum of positive solutions.
Quantitative stability for quasilinear parabolic equations
本文研究了准线性抛物型方程在扰动下的稳定性，通过比较论证方法建立了显式收敛速率，并对p-抛物方程的指数扰动和正则化近似提供了定量估计。
This article sets forth results on the existence, a priori estimates and boundedness of positive solutions of a singular quasilinear systems of elliptic equations involving variable exponents. The approach is based on Schauder's fixed point Theorem. A Moser iteration procedure is also obtained for singular cooperative systems involving variable exponents establishing a priori estimates and boundedness of solutions.
In this paper we develop a geometric theory for quasilinear parabolic problems in weighted $L_p$-spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the $\om$-limit sets and the long-time behaviour of the solutions. These techniques are applied to a free boundary value problem. The results in this paper are mainly based on maximal regularity tools in (weighted) $L_p$-spaces.
Recovering Initial States in Certain Quasilinear Parabolic Problems from Time Averages
本文研究从时间平均值重建拟线性抛物方程初始状态的反问题，在正则性和超线性增长条件下证明唯一恢复，并验证于趋化模型和反应-扩散系统。
We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-Δ_{p(x)}u = f(x,u)$ in $ (0,T)\timesΩ$; $u = 0$ on $(0,T)\times\partialΩ$; $u(0,x)=u_0(x)$ in $Ω$; involving the $p(x)$-Laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.
In this paper we prove finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. This is done in case of nonlinear diffusion and also in the case of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is assumed not to decay. Moreover, it is shown that the above-mentioned lack of non-decay assumption is essential with respect to keeping the dichotomy finite-time blowup against boundedness of solutions. Namely, we prove that without the non-decay assumption possible asymptotic behaviour…
In this paper, we prove global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions to the quasilinear parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=\operatorname{div}(F),$$ in a bounded domain $Ω\times (0,T)\subset\mathbb{R}^{N+1}$, under minimal regularity assumptions on the boundary of domain and on nonlinearity $A$. Then results yields existence of a solution to the Riccati type parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=|\nabla u|^q+\operatorname{div}(F)+μ,$$ where $q>1$ and $μ$ is a bounded Radon measure.
In this paper we study a rather wide class of quasilinear parabolic problems with nonlinear boundary condition and nonstandard growth terms. It includes the important case of equations with a $p(t,x)$-Laplacian. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems. Our results seem to be new even in the constant exponent case.
In this paper, we are interested in obtaining a unified approach for $C^{1,α}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being \[ u_t - \text{div} (|\nabla u|^{p-2} \nabla u) = 0. \] without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a delicate adaptation of the covering argument developed by E.~DiBenedetto and A.~Friedman.
We study unbounded (viscosity) supersolutions of the Evolutionary p-Laplace Equation in the slow diffusion case. The supersolutions fall into two widely different classes, depending on whether they are locally summable to the power p-2 or not. Also the Porous Medium Equation is studied.
This paper deals with the analysis of the internal control with constraint of positive kind of a parabolic PDE with nonlinear diffusion when the time horizon is large enough. The minimal controllability time will be strictly positive. We prove a global steady state constrained controllability result for a quasilinear parabolic with nonlinearity in the diffusion term. Then, under suitable dissipative assumption in the system and local controllability results, we conclude the result to any initial datum and any target trajectory.
We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $Ω\times(0,+\infty)$, where $Ω$ is a bounded domain. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.
Boundary value problems for some quasilinear parabolic equations under minimal conditions on the coefficients
研究拟线性抛物边值问题，给出保证Sobolev类解存在唯一性的尖锐条件，包括数据的光滑性、相容性及确保极值原理的不等式，证明基于先验估计和参数延拓法。
We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally "self-similar" in a special way. We prove the existence, uniqueness and asymptotic stability of an expanding solution which is self-similar at discrete times. We also study the existence and uniqueness of a shrinking solution which is self-similar at discrete times.
We study the Cauchy problem for fully nonlinear (stochastic) parabolic partial differential equations. We provide both in deterministic and stochastic case the existence of a maximal defined solution for the problem and we provide suitable blow-up criterion. The key idea is the use of the paradifferential operator calculus in order to reduce the fully nonlinear problem into an abstract quasilinear (stochastic) parabolic equation. This allows us to use some recent results on abstract quasilinear (stochastic) evolution equations in Banach spaces. To do so, we analyse the properties of the paradi…
This paper deals with the quasilinear parabolic-elliptic chemotaxis system with logistic source and nonlinear production, \begin{equation*} \begin{cases} u_t=\nabla \cdot (D(u) \nabla u) - \nabla \cdot (S(u)\nabla v) + λu - μu^κ, & x\inΩ,\ t>0, \\[1mm] 0=Δv - \overline{M_f}(t) + f(u), & x\inΩ,\ t>0, \end{cases} \end{equation*} where $λ>0$, $μ>0$, $κ>1$ and $\overline{M_f}(t):=\frac{1}{|Ω|}\int_Ω f(u(x,t))\,dx$, and $D$, $S$ and $f$ are functions generalizing the prototypes \begin{align*} D(u)=(u+1)^{m-1},\quad S(u)=u(u+1)^{α-1}\quad\mbox{and}\quad f(u)=u^\ell \end{align*} with $m\in\mathbb{R}$…
We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.
The global existence of weak solutions to a class of quasilinear parabolic equations with nonlinearities depending on first order terms and integrable data in a moving domain is investigated. The class includes the $p$-Laplace equation as a special case. Weak solutions are shown to be global by obtaining appropriate estimates on the gradient as well as a suitable version of Aubin-Lions lemma in moving domains.
In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation \[ u_t-\mathrm{div}\left(|\nabla u|^{p-2}\nabla u\right)=-|u|^{β-1}u+α|u|^{q-2}u, \] where $p>1,β>0$, $q\geq1$ and $α>0$. By using Gagliardo-Nirenberg type inequality, energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.
This paper is concerned with existence and uniqueness of solutions to two kinds of quasilinear parabolic equations. One is described as the form which includes the porous media and fast diffusion type equations. The other is the Cahn--Hilliard type system. The present paper applies Brézis theory directly to both equations and gives existence results for these two equations even if the domain is unbounded. Moreover, an error estimate is also proved via apriori estimates obtained directly.
We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in \cite{Zhong} of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\Hn$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of quasilinear pde's in $\Hn$ modelled on the equation $$\p_t u= \sum_{i=1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$
Simultaneous stable determination of quasilinear terms for parabolic equations
该研究针对抛物方程两类拟线性项的同时恢复反问题，结合线性化、奇异解构造等方法推导了Lipschitz与Hölder稳定性估计，可为数值重构收敛速率分析提供支撑。
A Sharp Mass Threshold for Boundedness in a Critical Quasilinear Chemotaxis Model With Indirect Signal Production: The Radially Symmetric Case
针对空间维数大于2的具有间接信号产生的临界拟线性趋化模型，在径向对称情形下确立了尖锐质量阈值M_c，明确了不同质量与能量条件下解的全局有界性或无穷时间爆破行为，并关联了其与相关不等式最优常数的关系。
Our study of abstract quasi-linear parabolic problems in time-weighted L_p-spaces, begun in [17], is extended in this paper to include singular lower order terms, while keeping low initial regularity. The results are applied to reaction-diffusion problems, including Maxwell-Stefan diffusion, and to geometric evolution equations like the surface-diffusion flow or the Willmore flow. The method presented here will be applicable to other parabolic systems, including free boundary problems.
We study the possibility of prescribing infinite initial values for solutions of the Evolutionary $p$-Laplace Equation in the fast diffusion case $p>2$. This expository note has been extracted from our previous work. When infinite values are prescribed on the whole initial surface, such solutions can exist only if the domain is a space-time cylinder.
We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal $L_p$-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier-Stokes problem, convection-diffusion equations,the Nernst-Planck-Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and som…
We investigate the asymptotic behavior of solutions for quasilinear parabolic equations in bounded intervals. In particular, we are concerned with a special class of solutions, called interface solutions, which exhibit e metastable behavior, meaning that their convergence towards the asymptotic configuration of the system is exponentially slow. The key of our analysis is a linearization around an approximation of the steady state of the problem, and the reduction of the dynamics to a one-dimensional motion, describing the slow convergence of the interfaces towards the equilibrium.
This note is devoted to a small, but essential, extension of Theorem 2.1 of our recent paper [6]. The improvement is explained in Section 1 and proved in Section 2. The importance of the extension is demonstrated in Section 3 with an application to the Navier-Stokes system in critical $L_q$-spaces.
In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the duality pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the compactness argument developed in \cite{bulicek2018well}. In order to obtain the a priori estimates, we make use of the full Calderón-Zygmund machinery developed in the past few years and combine it with some sharp bounds for the subclass of Muckenhoupt weights considered in this paper.
Linearized Stability of Non-Isolated Equilibria of Quasilinear Parabolic Problems in Interpolation Spaces
本文研究了准线性抛物型问题在插值空间中非孤立平衡解的稳定性，通过灵活的插值方法和低正则性假设，扩展了此前结果，并应用于毛细力驱动的Hele-Shaw问题和分数平均曲率流。
We consider a hyperbolic-parabolic model of vasculogenesis in the multidimensional case. For this system we show the global existence of smooth solutions to the Cauchy problem, using suitable energy estimates. Since this model does not enter in the classical framework of dissipative problems, we analyze it combining the features of the hyperbolic and the parabolic parts. Moreover we study the asymptotic behavior of those solutions showing their decay rates by means of detailed analysis of the Green function for the linearized problem.
Quasilinear parabolic equations with superlinear nonlinearities in critical spaces
本文在临界空间中研究具有超线性非线性的拟线性抛物方程的适定性，利用抛物正则化效应建立半流，并通过大气流动等例子验证结果。
We prove boundedness of gradients of solutions to quasilinear parabolic system, the main part of which is a generalization to p-Laplacian and its right hand side's growth depending on gradient is not slower (and generally strictly faster) than p - 1. This result may be seen as a generalization to the classical notion of a controllable growth of right hand side, introduced by Campanato, over gradients of p-Laplacian-like systems. Energy estimates and nonlinear iteration procedure of a Moser type are cornerstones of the used method.
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