In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) via these weights under an extra mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned two-weighted boundedness of the uncentered parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón--Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.
This work discusses parabolic Muckenhoupt weights on spaces of homogeneous type, i.e.\ quasi-metric spaces with both a doubling measure and an additional monotone geodesic property. The main results include a characterization in terms of weighted norm inequalities for parabolic maximal operators, a reverse Hölder inequality, and a Jones-type factorization result for this class of weights. The connection between the space of parabolic bounded mean oscillation and parabolic Muckenhoupt weights is studied by applying a parabolic John--Nirenberg lemma. A Coifman--Rochberg-type characterization of …
We investigate parabolic Muckenhoupt weights and functions of bounded mean oscillation (BMO) related to nonlinear parabolic partial differential equations. The main result gives a full characterization of weak and strong type weighted norm inequalities for parabolic forward in time maximal operators. In addition, we give a Jones type factorization result for the parabolic Muckenhoupt weights and a Coifman-Rochberg type characterization of the parabolic BMO from Moser's seminal paper through parabolic Muckenhoupt weights and maximal functions.
We derive weighted Sobolev-Poincaré type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class, so-called the parabolic $A_p$-class, where only the parabolic cubes are involved in the definition.
We present the unique solvability in Sobolev spaces of time fractional parabolic equations in divergence and non-divergence forms. The leading coefficients are merely measurable in $(t,x_1)$ for $a^{ij}$, $1 \leq i,j \leq d$, $(i,j) \neq (1,1)$. The coefficient $a^{11}$ is merely measurable locally either in $t$ or $x_1$. As functions of the remaining variables, the coefficients have small mean oscillations. We consider mixed norm Sobolev spaces with Muckenhoupt weights. Our results generalize previous work on parabolic equations with time fractional derivatives to a much larger class of coeff…
We prove weighted mixed-norm $L^q_t(W^{2,p}_x)$ and $L^q_t(C^{2,α}_x)$ estimates for $1<p,q<\infty$ and $0<α<1$, weighted mixed weak-type estimates for $q=1$, $L^\infty_{t}(L^p_x)-BMO_t(W^{2,p}_x)$, and $L^\infty_{t}(C^α_x)-BMO_t(C^{2,α}_x)$, and a.e.~pointwise formulas for derivatives, for solutions $u=u(t,x)$ to parabolic equations of the form $$\partial_tu-a^{ij}(t)\partial_{ij}u+u=f\quad t\in\mathbb{R},~x\in\mathbb{R}^n$$ and for the Cauchy problem $$\begin{cases} \partial_tv-a^{ij}(t)\partial_{ij}v+v=f&\hbox{for}~t >0,~x\in\mathbb{R}^n \\ v(0,x)=g&\hbox{for}~x\in\mathbb{R}^n. \end{cases}$…
We prove weighted $L_{p,q}$-estimates for divergence type higher order elliptic and parabolic systems with irregular coefficients on Reifenberg flat domains. In particular, in the parabolic case the coefficients do not have any regularity assumptions in the time variable. As functions of the spatial variables, the leading coefficients are permitted to have small mean oscillations. The weights are in the class of Muckenhoupt weights $A_p$. We also prove the solvability of the systems in weighted Sobolev spaces.
In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted $L_{q}$-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt $A_{\infty}$-class. In Besov space case we have the restriction that the microscopic parameter equals to $q$. Going beyond the $A_{p}$-range, where $p$ is the integrability paramet…
In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse Hölder's inequalities are established. Lipschitz estimates for weak solutions are proved for homogeneous equations with singular degenerate coefficients depending only on one spatial variable. These estimates are then used to establish interior, boundary, and global weighted estimates of Calderón-Zygmund type for weak solutions, assuming that the coefficients are partially VMO (vani…
We prove the boundedness of the non-local operator \[ \mathcal{L}^a u(x)=\int_{\mathbb{R}^d} \left(u(x+y)-u(x)-χ_α(y)\big(\nabla u(x),y\big)\right) a(x,y)\frac{dy}{|y|^{d+α}} \] from $H_{p,w}^α(\mathbb{R}^d)$ to $L_{p,w}(\mathbb{R}^d)$ for the whole range of $p \in (1,\infty)$, where $w$ is a Muckenhoupt weight. The coefficient $a(x,y)$ is bounded, merely measurable in $y$, and Hölder continuous in $x$ with an arbitrarily small exponent. We extend the previous results by removing the largeness assumption on $p$ as well as considering weighted spaces with Muckenhoupt weights. Using the boundedn…
In this paper, we obtain weighted norm inequalities for the spatial gradients of weak solutions to quasilinear parabolic equations with weights in the Muckenhoupt class $A_{\frac{q}{p}}(\mathbb{R}^{n+1})$ for $q\geq p$ on non-smooth domains. Here the quasilinear nonlinearity is modelled after the standard $p$-Laplacian operator. Until now, all the weighted estimates for the gradient were obtained only for exponents $q>p$. The results for exponents $q>p$ used the full complicated machinery of the Calderón-Zygmund theory developed over the past few decades, but the constants blow up as $q \right…
We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the equations under consideration appear in the study of fractional heat and fractional Laplace equations. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved under certain smallness conditions on the weighted mean oscillations of the coefficients in small parabolic cylinders. Our results are new even when the…
We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: \begin{equation*} \begin{cases} \partial_t \overline{u} - y^{-a} \nabla \cdot(y^a \nabla \overline{u}) = 0 \quad &\text{ in } \mathbb{B}_1^+ \times (-1,0) \\ -\partial_y^a \overline{u} = q(x,t)u \quad &\text{ on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where $a\in(-1,1)$, is a fixed parameter $\mathbb{B}_1^+\subset \mathbb{R}^{N+1}$ is the upper unit half ball and $B_1$ is the unit ball in $\mathbb{R}^N$. Our mai…
In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm Liouville problem in a general star graph with mixed Dirichlet and Ne…
In this paper, for a family of second-order parabolic equation with rapidly oscillating and time-dependent periodic coefficients, we are interested in an approximate two-sphere one-cylinder inequality for these solutions in parabolic periodic homogenization, which implies an approximate quantitative propagation of smallness. The proof relies on the asymptotic behavior of fundamental solutions and the Lagrange interpolation technique.
Let $v:[0,T]\times \R^d \to \R$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [σσ^\perp]_{il} \partial_{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition $g$, where the coefficients are time- and state-dependent, and satisfy certain regularity assumptions. Let $X=(X_t)_{t\in [0,T]}$ be the associated $\R^d$-valued diffusion process on some appropriate $(Ω,\cF,\Q)$. For $p\in [2,\infty)$ and a measure $d¶=λ_T d\Q$, where $λ_T$ satisfies the Muckenhoupt condition $A_α$ for $α\in (1,p)$, we relate the behavior of $\|g(X_T)-…
We prove L^$\infty$ estimates on the densities that are obtained via the JKO scheme for a general form of a parabolic-elliptic Keller-Segel type system, with arbitrary diffusion, arbitrary mass, and in arbitrary dimension. Of course, such an estimate blows up in finite time, a time proportional to the inverse of the initial L^$\infty$ norm. This estimate can be used to prove short-time well-posedness for a number of equations of this form regardless of the mass of the initial data. The time of existence of the constructed solutions coincides with the maximal time of existence of Lagrangian sol…
Extending work of Texier and Zumbrun in the semilinear non-re ection symmetric case, we study O(2) transverse Hopf bifurcation, or \cellular instability," of viscous shock waves in a channel, for a class of quasilinear hyperbolic{parabolic systems including the equations of thermoviscoelasticity. The main difficulties are to (i) obtain Fr'echet differentiability of the time-T solution operator by appropriate hyperbolic{parabolic energy estimates, and (ii) handle O(2) symmetry in the absence of either center manifold reduction (due to lack of spectral gap) or (due to nonstandard quasilinear hyp…
A finite element analysis of a Dirichlet boundary control problem governed by the linear parabolic equation is presented in this article. The Dirichlet control is considered in a closed and convex subset of the energy space $H^1(Ω\times(0,T)).$ We prove well-posedness and discuss some regularity results for the control problem. We derive the optimality system for the optimal control problem. The first order necessary optimality condition results in a simplified Signorini type problem for control variable. The space discretization of the state variable is done using conforming finite elements, …
We introduce a definition of the fractional Laplacian $(-Δ)^{s(\cdot)}$ with spatially variable order $s:Ω\to [0,1]$ and study the solvability of the associated Poisson problem on a bounded domain $Ω$. The initial motivation arises from the extension results of Caffarelli and Silvestre, and Stinga and Torrea; however the analytical tools and approaches developed here are new. For instance, in some cases we allow the variable order $s(\cdot)$ to attain the values $0$ and $1$ leading to a framework on weighted Sobolev spaces with non-Muckenhoupt weights. Initially, and under minimal assumptions,…
