We prove existence of weak solutions to doubly degenerate diffusion equations $$\dot{u}={\Delta }_p{u}^{m-1}+f(m,p\ge 2)$$ by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains $\Omega \subset {ℝ}^n$ with Dirichlet or Neumann boundary conditions. The function $f$ can be an inhomogeneity or a nonlinearity involving terms of the form $f\left(u\right)$ or $div\left(F\right(u\left)\right)$. In the appendix, an introduction to weak differentiability...

We study the existence of solutions of the nonlinear parabolic problem $\partial u/\partial t-div\left[\right|Du-\Theta \left(u\right){|}^{p-2}(Du-\Theta \left(u\right))]+\alpha \left(u\right)=f$ in ]0,T[ × Ω, $\left(\right|Du-\Theta \left(u\right){|}^{p-2}(Du-\Theta \left(u\right)))·\eta +\gamma \left(u\right)=g$ on ]0,T[ × ∂Ω, u(0,·) = u₀ in Ω, with initial data in L¹. We use a time discretization of the continuous problem by the Euler forward scheme.

The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form $$u_t=v^p\left(\Delta u+a{\int }_{\Omega }u\mathrm{d}x\right),v_t=u^q\left(\Delta v+b{\int }_{\Omega }v\mathrm{d}x\right)$$ with null Dirichlet boundary conditions. By using the standard approximation method, we first give a series of fine a priori estimates for the solution of the corresponding approximate problem. Then using the diagonal method, we get the local existence and the bounds of the solution $(u,v)$ to this problem. Moreover, a necessary and sufficient condition for the non-global existence...

In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.

In this paper we homogenize monotone parabolic problems with two spatial scales and any number of temporal scales. Under the assumption that the spatial and temporal scales are well-separated in the sense explained in the paper, we show that there is an H-limit defined by at most four distinct sets of local problems corresponding to slow temporal oscillations, slow resonant spatial and temporal oscillations (the “slow” self-similar case), rapid temporal oscillations, and rapid resonant spatial and...

This paper deals with a quasilinear parabolic-parabolic-elliptic attraction-repulsion chemotaxis system. Boundedness, stabilization and blow-up in this system of the fully parabolic and parabolic-elliptic-elliptic versions have already been proved. The purpose of this paper is to derive boundedness and stabilization in the parabolic-parabolic-elliptic version.

Large time behavior of the solution to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. Furthermore, the rate of convergence is given. Initial-boundary value problem with mixed boundary conditions is considered.

In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems.

We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal $L^p$-regularity of the periodic Laplace and Stokes operators and a local-in-time existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group $G:={ℝ}^{n-1}\times ℝ/L\mathbb{Z}$ to obtain an $ℛ$-bound for the...

This paper considers the initial-boundary value problem for the nonlinear diffusion equation with the perturbation term $$u_t+(-\Delta +1)\beta \left(u\right)+G\left(u\right)=g\text{in}\Omega \times (0,T)$$ in an unbounded domain $\Omega \subset {ℝ}^N$ with smooth bounded boundary, where $N\in \mathbb{N}$, $T>0$, $\beta$, is a single-valued maximal monotone function on $ℝ$, e.g., $$\beta \left(r\right)={\left|r\right|}^{q-1}r(q>0,q\ne 1)$$ and $G$ is a function on $ℝ$ which can be regarded as a Lipschitz continuous operator from ${\left(H^1\left(\Omega \right)\right)}^{*}$ to ${\left(H^1\left(\Omega \right)\right)}^{*}$. The present work establishes existence and estimates for the above problem.

In this paper, we consider the blow-up solutions for a quasilinear parabolic partial differential equation $u_t=u^2(u_{xx}+u)$. We numerically investigate the blow-up rates of these solutions by using a numerical method which is recently proposed by the authors [3].

We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data. The design of the function spaces is guided by tools and constructions from harmonic analysis, like maximal functions, square functions and Carleson measures. We construct solutions under virtually optimal scale invariant conditions on the initial data. Applications include BMO initial data for the harmonic map heat flow and the Ricci-DeTurck flow for initial metrics with small local...