我们证明了强退化各向异性抛物方程局部弱解的局部有界性。方程为∂_t u = Σ_{i=1}^n ∂_x_i[(|u_{x_i}| - δ_i)_+^{p-1} u_{x_i}/|u_{x_i}|]，其中Ω是R^n中的有界域，n≥2，p≥2，δ_1,…,δ_n是非负数，(·)_+表示正部。本文的主要创新在于上述方程结合了各向异性结构与强退化行为。本文的核心结果是将最初为抛物p-拉普拉斯证明的经典有界性定理扩展到广泛退化各向异性设置中。作为副产品，我们还获得了上述方程各向同性情形下局部弱解的局部有界性。
英文摘要：
We prove the local boundedness of local weak solutions to the parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-δ_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,Ω_{T}=Ω\times(0,T]\,, \] where $Ω$ is a bounded domain in $\mathbb{R}^{n}$ with $n\geq2$, $p\geq2$, $δ_{1},\ldots,δ_{n}$ are non-negative numbers and $\left(\,\cdot\,\right)_{+}$ denotes the positive part. The main novelty here is that the above equation combines an orthotropic structure with a strongly degenerate behavior. The core result of this paper thus extends a classical boundedness theorem, originally proved for the parabolic $p$-Laplacian, to a widely degenerate anisotropic setting. As a byproduct, we also obtain the local boundedness of local weak solutions to the isotropic counterpart of the above equation.
We consider a quasilinear degenerate parabolic equation driven by the orthotropic $p-$Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.
We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming the initial datum is localized with respect to a coordinate having slow diffusion rate, we bound the corresponding directional velocity of the support along the flow. The expansion rate is shown to be optimal for large times.
We prove the boundedness of the time derivative in the parabolic Signorini problem, as well as establish its Hölder continuity at regular free boundary points.
In the present note, we give a concise proof for the equivalence between the local boundedness property for parabolic Dirichlet BVP's and the gaussian upper bound for their Green functions. The parabolic equations we consider are of general divergence form and our proof is essentially based on the gaussian upper bound by Daners \cite{Da} and a Caccioppoli's type inequality. We also show how the same analysis enables us to get a weaker version of the local boundedness property for parabolic Neumann BVP's assuming that the corresponding Green functions satisfy a gaussian upper bound.
The purpose of this paper is to obtain an upper bound for the fundamental solution for parabolic Cauchy problem u'=Au, where A is a second order elliptic partial differential operator with unbounded coefficients such that its potential and the potential of the formal adjoint of the operator A are bounded from below.
In this thesis, a unified approach to prove the boundedness of gradients of solutions to degenerate and singular elliptic and parabolic phi-Laplacian systems is presented. At first, a Cacciopoli-type energy inequality with an additional function f which can be chosen freely is proven. Then, Di Giorgi's method is applied using level sets which will lead to L-infinity-estimates on the gradient of the weak solution.
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 $q < p\frac{N+1}{N-1}$.
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 a parabolic-parabolic Keller-Segel system of chemotaxis model with singular sensitivity $u_t=Δu-χ\nabla\cdot(\frac{u}{v}\nabla v)$, $v_t=kΔv-v+u$ under homogeneous Neumann boundary conditions in a smooth bounded domain $Ω\subset\mathbb{R}^n$ $(n\geq2)$, with $χ,k>0$. It is proved that for any $k>0$, the problem admits global classical solutions, whenever $χ\in\big(0,-\frac{k-1}{2}+\frac{1}{2}\sqrt{(k-1)^2+\frac{8k}{n}}\big)$. The global solutions are moreover globally bounded if $n\le 8$. This shows an exact way the size of the diffusion constant $k$ of the chemicals $v$ effects th…
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 the parabolic chemotaxis model \[ u_t=Δu - χ\nabla\cdot(\frac uv \nabla v), \qquad\qquad v_t=Δv - v + u\] in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for $χ\in(0,χ_0)$ for some $χ_0>1$, thereby proving that the value $χ=1$ is not critical in this regard. Our main tool is consideration of the energy functional \[ \mathcal{F}_{a,b}(u,v)=\int_Ωu\ln u - a \int_Ωu\ln v + b \int_Ω|\nabla \sqrt{v}|^2 \] for $a>0$, $b\geq 0$, where using nonzero values of $b$ appears to be new in this context.
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 prove local boundedness of variational solutions to the double phase equation \begin{align*} \partial_t u +& P.V.\int_{\mathbb{R}^N}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\ &+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qs'}} \,dy = 0, \end{align*} under the restrictions $s,s'\in (0,1),\, 1 < p \leq q \leq p\,\frac{2s+N}{N}$ and the non-negative function $(x,y)\mapsto a(x,y)$ is assumed to be measurable and bounded.
We study global-in-time well-posedness and the behaviour and of the solution to Cauchy problem in the classical Keller-Segel system with logistic term \begin{equation*} \left. \aligned \partial_tn-Δn=&-χ\nabla\cdot(n\nabla c)+\la n-μn^2 τ\partial_tc-Δc=&-c+n \endaligned \right\}\quad\text{in}\,\,\,\RR^d\times\RR^+, \end{equation*} where $d\ge 1$, $τ,\, χ,\, μ>0$ and $λ\ge 0$. It's inspired by a previous result \cite[M. Winkler, Commun. Part. Diff. Eq., 35 (2010), 1516-1537]{Win10}, where the global-in-time boundedness of the above Keller-Segel system in smooth \emph{bounded }convex domains is …
This paper deals with classical solutions to the parabolic-parabolic system \begin{align*} \begin{cases} u_t=Δ(γ(v) u ) &\mathrm{in}\ Ω\times(0,\infty), \\[1mm] v_t=Δv - v + u &\mathrm{in}\ Ω\times(0,\infty), \\[1mm] \displaystyle \frac{\partial u}{\partial ν} = \frac{\partial v}{\partial ν} = 0 &\mathrm{on}\ \partialΩ\times (0,\infty), \\[1mm] u(\cdot,0)=u_0, \ v(\cdot,0)=v_0 &\mathrm{in}\ Ω, \end{cases} \end{align*} where $Ω$ is a smooth bounded domain in $\mathbf{R}^n$($n \geq 3$), $γ(v)=v^{-k}$ ($k>0$) and the initial data $(u_0,v_0)$ is positive and regular. This system has striking featu…
In this paper we consider the initial Neumann boundary value problem for a degenerate Keller--Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos--Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the moti…
This paper deals with the fully parabolic attraction-repulsion chemotaxis system with signal-dependent sensitivities, \begin{align*} \begin{cases} u_t=Δu-\nabla \cdot (uχ(v)\nabla v) +\nabla \cdot (uξ(w)\nabla w), &x \in Ω,\ t>0,\\[1.05mm] v_t=Δv-v+u, &x \in Ω,\ t>0,\\[1.05mm] w_t=Δw-w+u, &x \in Ω,\ t>0 \end{cases} \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \ge 2)$ is a bounded domain with smooth boundary, $χ, ξ$ are functions satisfying some conditions. Global existence and boundedness of classical solutions to the syst…
This paper is concerned with global solvability of a fully parabolic system of Keller--Segel-type involving non-monotonic signal-dependent motility. First, we prove global existence of classical solutions to our problem with generic positive motility function under a certain smallness assumption at infinity, which however permits the motility function to be arbitrarily large within a finite region. Then uniform-in-time boundedness of classical solutions is established whenever the motility function has strictly positive lower and upper bounds in any dimension $N\geq1$, or decays at a certain s…
This paper is concerned with a parabolic-elliptic Keller-Segel system where both diffusive and chemotactic coefficients (motility functions) depend on the chemical signal density. This system was originally proposed by Keller and Segel in \cite{KS-1971-JTB2} to describe the aggregation phase of {\it Dictyostelium discoideum} cells in response to the secreted chemical signal cyclic adenosine monophosphate (cAMP), but the available analytical results are very limited by far. Considering system in a bounded smooth domain with Neumann boundary conditions, we establish the global boundedness of sol…
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 the following fully parabolic Keller--Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=Δu-χ\nabla\cdot(u\nabla v)+ au-μu^2,\quad x\in Ω, t>0, \disp{v_t=Δv- v +u},\quad x\in Ω, t>0, \end{array}\right.\eqno(KS) $$ over a bounded domain $Ω\subset\mathbb{R}^N(N\geq1)$, with smooth boundary $\partialΩ$, the parameters $a\in \mathbb{R}, μ>0, χ>0$. It is proved that if $μ>0$, then $(KS)$ admits a global weak solution, while if $μ>\frac{(N-2)_{+}}{N}χC^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1}$, then $(KS)$ possesses a global classical solution which is bounded, where $C^{…
The current paper considers the boundedness of solutions to the following quasilinear Keller-Segel model (with logistic source) $$\left\{\begin{array}{ll} u_t = \nabla\cdot(D(u)\nabla u)-χ\nabla\cdot(u\nabla v)+μ(u-u^2),\quad x\in Ω, t>0,\\ v_t-Δv = u-v,\quad x\in Ω, t>0,\\ (D(u)\nabla u-χu\cdot \nabla v)\cdot ν= \frac{\partial v}{\partialν}=0,\quad x\in \partialΩ, t>0,\\ u(x,0) = u_0(x),\quad v(x,0) = v_0(x),\ \ x\in Ω, \end{array}\right.$$ where $Ω\subset\mathbb{R}^N(N\geq1)$ is a bounded domain with smooth boundary $\partialΩ,$ $χ>0$ and $μ\geq0$. We prove that for nonnegative and suitably …
An Explicit Logistic Damping Criterion for Boundedness in a Fully Parabolic Keller--Segel System
Wuhan Institute of Physics and Mathematics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences（中国科学院精密测量科学与技术创新研究院武汉物理与数学研究所）
This paper deals with the two-species chemotaxis-competition system $u_t = d_1 Δu - χ_1 \nabla \cdot (u \nabla w) + μ_1 u(1 - u - a_1 v)$, $v_t = d_2 Δv - χ_2 \nabla \cdot (v \nabla w) + μ_2 v(1 - a_2 u - v)$, $0 = d_3 Δw + αu + βv - γw$, where $Ω$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $n\ge 2$; $χ_i$ and $μ_i$ are constants satisfying some conditions. The above system was studied in the cases that $a_1,a_2\in (0,1)$ and $a_1>1>a_2$, and it was proved that global existence and asymptotic stability hold when $\frac{χ_i}{μ_i}$ are small. However, the conditions in the above…
This paper deals with the quasilinear fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=\nabla \cdot (D(u)\nabla u) -\nabla \cdot (G(u)χ(v)\nabla v) +\nabla\cdot(H(u)ξ(w)\nabla w), \quad v_t=d_1Δv+αu-βv, \quad w_t=d_2Δw+γu-δw, \quad x \in Ω,\ t>0, \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω\subset \mathbb{R}^n$ $(n \ge 1)$ is a bounded domain with smooth boundary, $d_1, d_2, α, β, γ, δ>0$ are constants. Also, the diffusivity $D$, the density-dependent sensitivities $G, H$ fulfill $D(s)=a_0(s+1)^{m-1}$ with $a_0>0$ and $m \…
This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -χu(u+1)^{p-2}\nabla v +ξu(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=Δv+αu-βv, \\[1.05mm] 0=Δw+γu-δw \end{cases} \end{align*} in a bounded domain $Ω\subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partialΩ$, where $m, p, q \in \mathbb{R}$, $χ, ξ, α, β, γ, δ>0$ are constants. Moreover, it is supposed that the function $f$ satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when considering blow-up, it is assumed that $m>0…
This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic type $ u_t=\nabla\cdot(ϕ(u)\nabla u)-χ\nabla\cdot(u\nabla v)+g(u)$, $τv_t=Δv-v+u$ in $Ω\times (0,T)$, subject to nonnegative initial data and homogeneous Neumann boundary condition, where $Ω$ is smooth and bounded domain in $\mathbb{R}^n$, $n\ge 2$, $ϕ$ and $g$ are smooth and positive functions satisfying $ks^p\leϕ$ when $s\ge s_0>1$, $g(s) \le as - μs^2$ for $s>0$ with $g(0)\ge0$ and constants $a\ge 0$, $τ,χ,μ>0$. It was known that the model without the logistic s…
For the time-space fractional degenerate Keller-Segel equation \begin{equation*} \begin{cases} \partial _{t}^{β}u=-(-Δ)^{\fracα{2}}(ρ(v)u),& t>0\\ (-Δ)^{\fracα{2}} v+v=u,& t>0 \end{cases} \end{equation*} $x\inΩ, Ω\subset \mathbb{R}^{n}, β\in (0,1),α\in (1,2)$, we consider for $n\geq 3$ the problem of finding a time-independent upper bound of the classical solution such that as $θ>0,C>0$ \begin{equation*} \left \| u(\cdot ,t)-\overline{u_{0}} \right \|_{L^{\infty }(Ω)}+\left \| v(\cdot ,t)-\overline{u_{0}} \right \|_{W^{1,\infty }(Ω)}\leq Ce^{(-θ)^{1/β}t}, \end{equation*} where $\overline{u_{0}…
This paper deals with the Neumann boundary value problem for the system $$u_t=\nabla\cdot\left(D(u)\nabla u\right)-\nabla\cdot\left(S(u)\nabla v\right)+f(u) ,\quad x\inΩ,\ t>0$$ $$v_t=Δv-v+u,\quad x\inΩ,\ t>0$$ in a smooth bounded domain $Ω\subset\mathbb{R}^n$ $(n\geq1)$, where the functions $D(u)$ and $S(u)$ are supposed to be smooth satisfying $D(u)\geq Mu^{-α}$ and $S(u)\leq Mu^β$ with $M>0$, $α\in\mathbb{R}$ and $β\in\mathbb{R}$ for all $u\geq1$, and the logistic source $f(u)$ is smooth fulfilling $f(0)\geq0$ as well as $f(u)\leq a-μu^γ$ with $a\geq0$, $μ>0$ and $γ\geq1$ for all $u\geq0$. …
This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{equation*} u_t=Δu - \nabla \cdot (u χ(v)\nabla v), \quad v_t=Δv + u - v, \quad x\inΩ,\ t>0, \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$; $χ$ is a function satisfying $χ(s)\leq K(a+s)^{-k}$ for some $k\geq 1$ and $a\geq 0$. In the case that $k=1$, Fujie (J. Math. Anal. Appl.; 2015; 424; 675--684) established global existence of bounded solutions under the condition $K<\sqrt{\frac{2}{n}}$. On the other hand, when $k>1$, Winkler (Math. Nachr.; 2010; 283; 1664--1673) asserted glob…
In this paper, we study the following chemotaxis--haptotaxis system with (generalized) logistic source $$ \left\{\begin{array}{ll} u_t=Δu-χ\nabla\cdot(u\nabla v)- ξ\nabla\cdot(u\nabla w)+u(a-μu^{r-1}-w), \displaystyle{v_t=Δv- v +u},\quad \\ \displaystyle{w_t=- vw},\quad\\ \displaystyle{\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=\frac{\partial w}{\partial ν}=0},\quad x\in \partialΩ, t>0,\\ \displaystyle{u(x,0)=u_0(x)},v(x,0)=v_0(x),w(x,0)=w_0(x),\quad x\in Ω, \end{array}\right.\eqno(0.1) $$ %under homogeneous Neumann boundary conditions in a smooth bounded domain $\mathbb{R}^N(…
The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.
Here we describe a simple and fundamental approach to the maximal L^p regularity of parabolic equations, which only uses the concept of singular integrals of Volterra type. Knowledge of analytic semigroups, R-boundedness or H^\infty-functional calculus are not required.
This work deals with a parabolic chemotaxis model with nonlinear diffusion and nonlocal reaction source. The problem is formulated on the whole space and, depending on a specific interplay between the coefficients associated to such diffusion and reaction, we establish that all given solutions are uniformly bounded in time.
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
Global existence and boundedness of classical solutions are shown for a parabolic-elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of $L^\infty$-estimates on the second component of the system and is achieved by various comparison arguments.
