Alexander V. Romanov
We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have asymptotically finite-dimensional dynamics in the corresponding state space. This example is more natural in the class of evolutionary equations of parabolic type than those known earlier.
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The paper gives a comprehensive study of Inertial Manifolds for hyperbolic relaxations of an abstract semilinear parabolic equation in a Hilbert space. A new scheme of constructing Inertial Manifolds for such type of problems is suggested and optimal spectral gap conditions which guarantee their existence are established. Moreover, the dependence of the constructed manifolds on the relaxation parameter in the case of the parabolic singular limit is also studied.
We consider inertial manifolds and their approximation for a class of partial differential equations with a nonlocal Laplacian operator $-(-Δ)^{\fracα{2}}$, with $0<α<2$. The nonlocal or fractional Laplacian operator represents an anomalous diffusion effect. We first establish the existence of an inertial manifold and highlight the influence of the parameter $α$. Then we approximate the inertial manifold when a small normal diffusion $\varepsilon Δ$ (with $\varepsilon \in (0, 1)$) enters the system, and obtain the estimate for the Hausdorff semi-distance between the inertial manifolds with and…
In this paper we deal with a singular nonlocal phase field system with inertial term. The system has the logarithm of the absolute temperature $θ$ under time derivative. Although the system has a difficult mathematical point caused by the combination of $(\ln θ)_{t}$, the inertial term and the nonlocal diffusion term for the order parameter $φ$ (see Section 1.1), we can establish existence of solutions by a key estimate (see Remark 1.1).
We prove global existence and blow-up of solutions of initial-boundary value problem for nonlinear nonlocal parabolic equation with nonlinear nonlocal boundary condition. Obtained results depend on the behavior of variable coefficients for large values of time.
In this paper we consider initial boundary value problem for nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition and nonnegative initial datum. We prove comparison principle, global existence and blow-up of solutions.
In this paper we prove the optimal $L^p$-solvability of nonlocal parabolic equation with spatial dependent and non-smooth kernels.
We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a "Newtonian" nonlocal term.
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.
We establish Harnack's estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation. All results presented in this paper are provided together with quantitative estimates.
We survey recent regularity results for parabolic equations involving nonlocal operators like the fractional Laplacian. We extend the results of Felsinger-Kassmann (2013) and obtain regularity estimates for nonlocal operators with kernels not being absolutely continuous with respect to the Lebesgue measure.
In this paper we prove the existence of smooth solutions to fully nonlinear and nonlocal parabolic equations with critical index. The proof relies on the apriori Hölder estimate for advection fractional-diffusion equation established by Silvestre \cite{Si2}.
In this paper we consider an initial boundary value problem for a semilinear parabolic equation with nonlinear nonlocal boundary condition. We prove comparison principle, the existence theorem of a local solution and study the problem of uniqueness and nonuniqueness
This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions.
Some systems of parabolic equations with nonlocal initial conditions are studied. The systems arise when considering steady-state solutions to diffusive age-structured cooperative or competing species. Local and global bifurcation techniques are employed to derive a detailed description of the structure of positive coexistence solutions.
In this paper we consider an initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal Neumann boundary condition. We prove comparison principle, the existence theorem of a local solution and study the problem of uniqueness and nonuniqueness.
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.
We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calderón problems, resp…
This article deals with a nonlocal Penrose-Fife type phase field system with inertial term. We do not know whether we can prove existence of solutions in reference to Colli--Grasselli--Ito [Electron. J. Differential Equations 2002, No. 100, 32 pp.] or not (see Remark 1.1). In this paper we introduce a time discretization scheme (see Section 2), pass to the limit as the time step $h$ goes to $0$ and obtain an error estimate for the difference between continuous and discrete solutions (see Section 5).
Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.
We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on $C^m$-regularity of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.
Higher Hölder regularity for nonlocal parabolic equations with irregular kernels
本文研究了具有不规则核系数的非局部抛物型方程，在对非齐次项施加适当高阶可积性且对核系数作最小正则性假设的条件下，建立了该方程的高阶Hölder正则性。
We give a unified proof of Hölder regularity of weak solutions for mixed local and nonlocal $p$-Laplace type parabolic equations with the full range of exponents $1<p<\infty$. Our proof is based on the expansion of positivity together with the energy estimate and De Giorgi type lemma.
The paper is devoted to a comprehensive study of smoothness of inertial manifolds for abstract semilinear parabolic problems. It is well known that in general we cannot expect more than $C^{1,\varepsilon}$-regularity for such manifolds (for some positive, but small $\varepsilon$). Nevertheless, as shown in the paper, under the natural assumptions, the obstacles to the existence of a $C^n$-smooth inertial manifold (where $n\in\mathbb N$ is any given number) can be removed by increasing the dimension and by modifying properly the nonlinearity outside of the global attractor (or even outside the …
Blow-up of solutions for a semilinear parabolic equation with nonlinear memory and absorption under nonlinear nonlocal boundary condition
研究含非线性记忆与吸收项的半线性抛物方程在非线性非局部边界条件下的初边值问题，通过理论分析证明了解的整体存在性与有限时刻爆破。
This paper proves Hölder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first shows that viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative are Hölder continuous.
The fractional heat operator $(\partial_t-Δ_x)^s$ and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to $(\partial_t-Δ_x)^s$ and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator.
Optimal in tail Hölder estimates for weak solutions of the nonlocal parabolic p-Laplace equations on the Heisenberg group
本文研究了海森堡群上非局部抛物p-拉普拉斯方程弱解的最优尾Hölder估计问题，通过引入最优尾条件证明了弱解的Hölder连续性。
In this paper, we study the solvability of a Cauchy- Dirichlet problem for nonlinear parabolic equation with non standard growths and nonlocal terms. We show the existence of weak solutions of the considered problem under more general conditions. In addition, we obtain some results on the behavior of the solution when the problem is homogeneous.
Nonlocal to local convergence of phase field systems with inertial term
本文研究带惯性项和非局部卷积算子的双曲相场系统，严格证明当逼近参数趋于零时其渐近收敛于带椭圆Laplacian算子的局部系统，分析方法借鉴了Cahn--Hilliard模型族的非局部到局部收敛性质。
On the modulus of continuity of solutions to nonlocal parabolic equations
本文在极小及稍强尾部条件下，分别量化并推导了非局部抛物型方程局部有界弱解的一般连续性模与Hölder连续性模，并在具有可测核的非局部$p$-Laplacian框架下证明了这些正则性估计。
Gradient estimates for parabolic nonlinear nonlocal equations
本文建立了一类抛物型非线性非局部测度数据问题解的逐点梯度估计，并证明其解的一阶正则性及齐次方程解的Hölder连续空间梯度。
We establish a local boundedness estimate for weak subsolutions to a doubly nonlinear parabolic fractional $p$-Laplace equation. Our argument relies on energy estimates and a parabolic nonlocal version of De Giorgi's method. Furthermore, by means of a new algebraic inequality, we show that positive weak supersolutions satisfy a reverse Hölder inequality. Finally, we also prove a logarithmic decay estimate for positive supersolutions.
Mixed Local-Nonlocal Parabolic Equations in Heisenberg Group : Harnack Inequality with an Optimal Tail
本文通过正性扩展方法和聚类引理，证明了海森堡群上线性混合局部-非局部抛物方程具有最优尾部的哈纳克不等式，避免了Moser迭代和对数估计。
In this paper we study one dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. We establish global existence-uniqueness of classical solutions assuming that the initial-boundary data are sufficiently smooth and satisfy some compatibility conditions. Our approach is based on analysis of an equivalent system of nonlinear integral equations.
Hölder Continuity for Fully Fractional Parabolic Equations with Space-time Nonlocal Operators
本文研究含空间分数阶扩散与Marchaud型分数阶时间导数的全分数阶抛物型方程，在不假设解和非齐次项有界的条件下，引入时间依赖非局部尾项，利用非局部De Giorgi迭代技术证明弱解的局部有界性与局部Hölder连续性。
We prove Hölder regularity results for a class of nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel. Existence of weak solutions via approximating solutions is proved. Moreover, the Hölder continuity of viscosity solutions is obtained. We get the similar results as those obtained by Allen (see {\url{https://arxiv.org/abs/1610.10073}})
In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems.
Harnack inequality for doubly nonlinear mixed local and nonlocal parabolic equations
本文针对双重非线性混合局部与非局部抛物型方程，利用比较原理、局部有界性估计和正性扩张方法，建立了非负弱解的Harnack不等式。
Controllability issues for parabolic-elliptic systems involving nonlocal couplings
本文研究含非局部积分耦合的偏微分方程系统，分别证明非线性抛物-椭圆系统可借助 Carleman 估计实现控制，并揭示一维线性非局部抛物-抛物系统的可控性依赖耦合结构且可能失效。
In this paper, we consider the solution of a nonlocal parabolic equation. Focusing on the solutions with initial data at high energy level, we find the criteria for global existence and finite time blow up for the corresponding solution respectively. Moreover, we prove that there always exists blow up solution with negative Nehari functional no matter how large the energy is.
We investigate the mixed local and nonlocal parabolic $p$-Laplace equation \begin{align*} \partial_t u(x,t)-Δ_p u(x,t)+\mathcal{L}u(x,t)=0, \end{align*} where $Δ_p$ is the local $p$-Laplace operator and $\mathcal{L}$ is the nonlocal $p$-Laplace operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi-Nash-Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.
Global solutions to semilinear parabolic equations driven by mixed local-nonlocal operators
研究混合局部-非局部算子驱动的半线性抛物方程柯西问题，证明Fujita现象成立且临界值与分数阶拉普拉斯算子相同。
We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional $p$-Laplace equations which includes the fractional parabolic $p$-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results …
The paper is concerned with a system consisting of two coupled nonlinear parabolic equations with a cross-diffusion term, where the solutions at positive times define the initial states. The equations arise as steady state equations of an age-structured predator-prey system with spatial dispersion. Based on unilateral global bifurcation methods for Fredholm operators and on maximal regularity for parabolic equations, global bifurcation of positive solutions is derived.
Calderón-Zygmund theory of nonlocal parabolic equations with discontinuous coefficients
本文在核系数极小正则性条件下建立了非齐次非局部抛物型方程弱解的Calderón-Zygmund型估计，揭示了不连续系数情形下解具有超出局部方程对应情形的分数阶可微性提升。
We study a nonlocal evolution equation that involves a pseudo-parabolic third-order term. The equation models almost uni-directional two-phase flow in Brinkman regimes. We prove the existence of weak solutions for this equation. We also give a series of numerical examples that demonstrate the ability of the equation to support overshooting and explore the behavior of solutions in various limit regimes.
Maximal $L_q$-regularity of nonlocal parabolic equations in higher order Bessel potential spaces
本文研究了变系数分数阶抛物型方程，在任意非负阶Bessel位势空间中建立了极大$L_q$-正则性，并应用于证明分数阶多孔介质方程与非局部Kirchhoff方程的高阶正则性和瞬时平滑性。
Weak solutions to parabolic integro-differential operators of order $α\in (α_0, 2)$ are studied. Local a priori estimates of Hölder norms and a weak Harnack inequality are proved. These results are robust with respect to $α\nearrow 2$. In this sense, the presentation is an extension of Moser's result in 1971.
In this paper we prove existence and uniqueness of solutions to a nonlocal parabolic problem which generalizes the electric heating problem of a conducting body.
We obtain Schauder estimates for a class of concave fully nonlinear nonlocal parabolic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels. As a application, we prove that the solution to a translation invariant equation with merely bounded data is $C^σ$ in $x$ variable and $Λ^1$ in $t$ variable, where $Λ^1$ is the Zygmund space.
Nonlocal elliptic and parabolic equations with general stable operators in weighted Sobolev spaces
本文在加权Sobolev空间中研究$C^{1,τ}$开集上由对称稳定Lévy过程生成的非局部椭圆与抛物方程，允许Lévy测度高度奇异且时间变量仅可测，拓展了此类方程的可解性理论。
We consider a nonlocal parabolic model for a micro-electro-mechanical system. Specifically, for a radially symmetric problem with monotonic initial data, it is shown that the solution quenches, so that touchdown occurs in the device, in a situation where there is no steady state. It is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained. Numerical simulations illustrating the results are given.
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.
Dirichlet heat kernel estimates for parabolic nonlocal equations
本文研究了非局部抛物方程在C^{1,α}域内的边界正则性及Harnack原理，提出了适用于广泛非局部算子的估计方法，并获得了具有时间依赖系数的Dirichlet热核的精确双侧估计。
Non-autonomous problem for a $2m$-th order semilinear nonlocal parabolic equation
针对 $2m$ 阶非自治拟线性抛物方程，在非线性函数的增长和正则性条件下证明拉回吸引子的存在性与刻画，并讨论其自治版本。
This paper is concerned with a general class of fully nonlinear parabolic equations with monotone nonlocal terms. We investigate the quasiconvexity preserving property of positive, spatially coercive viscosity solutions. We prove that if the initial value is quasiconvex, the viscosity solution to the Cauchy problem stays quasiconvex in space for all time. Our proof can be regarded as a limit version of that for power convexity preservation as the exponent tends to infinity. We also present several concrete examples to show applications of our result.
We study local boundedness and Hölder continuity of a parabolic equation involving the fractional $p$-Laplacian of order $s$, with $0<s<1$, $2\leq p < \infty$, with a general right hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right hand side.
Inverse problem for multi-term fractional parabolic equation in two dimensional space, involving m + 1 Caputo fractional derivatives in time, is investigated. Presence of nonlocal boundary conditions leads to a non-self-adjoint spectral problem. A bi-orthogonal system of functions is used to construct the solution that involves double infinite series. Properties of multinomial Mittag-Leffler function and eigenfunctions are used to prove the classical nature of the solution under certain regularity conditions on the given datum.
A priori estimates of solutions to local and nonlocal superlinear parabolic problems
本文研究超线性抛物型问题变号解的先验估计，涵盖局部与非局部情形，应用包括爆破速率等，方法基于能量、插值及Pohozaev恒等式。
Symmetrization results for general nonlocal linear ellipitic and parabolic problems
该研究针对带齐次Dirichlet边界的一般非局部线性椭圆问题建立Talenti型质量集中对称化结果，将经典分数阶Laplacian情形纳入特例并推广已有结论，还通过隐式时间离散得到非局部抛物问题的对应结果。
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