Nonlocal Parabolic Equation Research Articles

Fujita exponent for non-local parabolic equation involving the Hardy–Leray potential

In this paper, we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy–Leray-type potential. More precisely, we consider the problem (wt-Δw)s=λ|x|2sw+wp+f,inRN×(0,+∞),w(x,t)=0,inRN×(-∞,0],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} (w_t-\\Delta w)^s=\\frac{\\lambda }{|x|^{2s}} w+w^p +f, &{}\\quad \ ext {in }\\mathbb {R}^N\ imes (0,+\\infty ),\\\\ w(x,t)=0, &{}\\quad \ ext {in }\\mathbb {R}^N\ imes (-\\infty ,0], \\end{array}\\right. } \\end{aligned}$$\\end{document}where N>2s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N> 2s$$\\end{document}, 0<s<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<s<1$$\\end{document} and 0<λ<ΛN,s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\lambda <\\Lambda _{N,s}$$\\end{document}, the optimal constant in the fractional Hardy–Leray inequality. In particular, we show the existence of a critical existence exponent p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_{+}(\\lambda , s)$$\\end{document} and of a Fujita-type exponent F(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(\\lambda ,s)$$\\end{document} such that the following holds:Let p>p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>p_+(\\lambda ,s)$$\\end{document}. Then there are not any non-negative supersolutions.Let p<p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p<p_+(\\lambda ,s)$$\\end{document}. Then there exist local solutions, while concerning global solutions we need to distinguish two cases:Let 1<p≤F(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 1< p\\le F(\\lambda ,s)$$\\end{document}. Here we show that a weighted norm of any positive solution blows up in finite time.Let F(λ,s)<p<p+(λ,s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(\\lambda ,s)<p<p_+(\\lambda ,s)$$\\end{document}. Here we prove the existence of global solutions under suitable hypotheses.

Qualitative properties of fractional convolution elliptic and parabolic operators in Besov spaces

The maximal Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ B_{p,q}^{s}$$\\end{document} and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of linear problem, the existence, uniqueness of maximal regular solution of corresponding fractional nonlinear equation is established.

Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem

We prove the local Hölder regularity of weak solutions to the mixed local nonlocal parabolic equation of the formut−Δu+P.V.∫Rnu(x,t)−u(y,t)|x−y|n+2sdy=0, where 0<s<1; for every exponent α0∈(0,1). Here, Δ is the usual Laplace operator. Next, we show that the gradients of weak solutions are also α-Hölder continuous for some α∈(0,1). Our approach is purely analytic and it is based on perturbation techniques.

Mixed local and nonlocal parabolic equation: Global existence, decay and blow-up

Mixed local and nonlocal parabolic equation: Global existence, decay and blow-up

On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution

&lt;abstract&gt;&lt;p&gt;In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.&lt;/p&gt;&lt;/abstract&gt;

Asymptotic behavior and blow-up of solutions for a nonlocal parabolic equation with a special diffusion process

Asymptotic behavior and blow-up of solutions for a nonlocal parabolic equation with a special diffusion process

Regularity of weak solutions for mixed local and nonlocal double phase parabolic equations

We study the mixed local and nonlocal double phase parabolic equation∂tu(x,t)−div(a(x,t)|∇u|q−2∇u)+Lu(x,t)=0 in QT=Ω×(0,T), where L is the nonlocal p-Laplace type operator. The local boundedness of weak solutions is proved by means of the De Giorgi-Nash-Moser iteration with the nonnegative coefficient function a(x,t) being bounded. In addition, when a(x,t) is Hölder continuous, we discuss the pointwise behavior and lower semicontinuity of weak supersolutions based on energy estimates and De Giorgi type lemma. Analogously, the corresponding results are also valid for weak subsolutions.

Application of homogenization and large deviations to a nonlocal parabolic semi-linear equation

&#x0D; &#x0D; &#x0D; We study the behavior of the solution for a class of nonlocal partial differential equation of parabolic-type with non-constant coefficients varying over length scale δ and nonlinear reaction term of scale 1/ε, related to stochastic differential equations driven by multiplicative isotropic α-stable Lévy noise (1 &lt; α &lt; 2). The behavior is required as ε tends to 0 with δ small compared to ε. Our homogenization method&#x0D; &#x0D; &#x0D;

Global existence, blow-up and dynamical behavior in a nonlocal parabolic problem with variational structure

In this paper, we study an initial boundary value problem for a nonlocal parabolic equation that includes a diffusion term and convex–concave nonlinearities. By establishing the Lq-estimate and analyzing its energy, we prove the existence of global solutions and obtain some blow-up conditions. Given the variational structure of the problem, we utilize the Mountain-pass theorem to demonstrate the existence of nontrivial steady-state solutions. Additionally, we establish the dynamical behavior of global solutions whose trajectory is relatively compact in H01(Ω), and which uniformly converge to some non-zero steady state after a long time, owing to the energy functional satisfying the P.S. condition. Finally, we derive an unstable steady states sequence by using another minimax theorem.

Global existence and blow-up of solutions of nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition

Global existence and blow-up of solutions of nonlinear nonlocal parabolic equation with absorption under nonlinear nonlocal boundary condition

Global Existence and Blow-up Solutions for a Parabolic Equation with Critical Nonlocal Interactions

AbstractIn this paper, we study the initial boundary value problem for the nonlocal parabolic equation with the Hardy–Littlewood–Sobolev critical exponent on a bounded domain. We are concerned with the long time behaviors of solutions when the initial energy is low, critical or high. More precisely, by using the modified potential well method, we obtain global existence and blow-up of solutions when the initial energy is low or critical, and it is proved that the global solutions are classical. Moreover, we obtain an upper bound of blow-up time for $$J_{\mu }(u_{0})&lt;0$$ J μ ( u 0 ) &lt; 0 and decay rate of $$H^{1}_{0}$$ H 0 1 and $$L^{2}$$ L 2 -norm of the global solutions. When the initial energy is high, we derive some sufficient conditions for global existence and blow-up of solutions. In addition, we are going to consider the asymptotic behavior of global solutions, which is similar to the Palais-Smale (PS for short) sequence of stationary equation.

Weak Harnack inequality for a mixed local and nonlocal parabolic equation

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-Giusti 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.

Galerkin stability of pullback attractors for nonautonomous nonlocal equations

We study Galerkin approximation of pullback attractors for a nonautonomous nonlocal parabolic equation. We show that the $ n $th Galerkin approximation system has a pullback attractor in the $ n $-dimensional subspace of the Lebesgue space. We then prove that the sequence of Galerkin pullback attractors is uniformly backward bounded and that the sequence of Galerkin solutions converges uniformly on any bounded set. Using these results, we establish upper semi-convergence of the sequence of Galerkin pullback attractors towards the pullback attractors of the original infinite-dimensional system.

Global dynamics of a nonlocal non-uniformly parabolic equation arising from the curvature flow* *The first author is supported by NSF of China (No. 12271437 and No. 12071203). The second author is supported by NSF of China (No. 11971498) and NSF of Guangdong Province (No. 2019A1515011339). The third author is supported by NSF of China (No. 11871148).

This paper studies a type of non-uniformly parabolic problem with nonlocal term ut=up(uxx+u−u¯)0<t<Tmax,0<x<a,ux(t,0)=ux(t,a)=00<t<Tmax,u(0,x)=u0(x)0<x<a, where p > 1, a > 0. First the classification of the finite-time blow-up/global existence phenomena based on the associated energy functional and explicit expression of all nonnegative steady states are demonstrated. More importantly, we derive that any bounded solution converges to some steady state as t → +∞. The difficulties in proving this convergence result lie in the existence of a continuum of steady states and the lack of the comparison principle due to the introduction of nonlocal term. To conquer these difficulties, we combine the applications of Lojasiewicz–Simon inequality and energy estimates.

Blow-Up Solution in Logarithmic Sensitivity and Indirect Signal Generation Keller–Segel System

In this paper, we consider the following indirect signal generation and logarithmic sensitivity ε n t = Δ n − χ ∇ · n ∇ ln c x ∈ Ω , t &gt; 0 c t = Δ c − c + w x ∈ Ω , t &gt; 0 w t = Δ w − w + n x ∈ Ω , t &gt; 0 under homogeneous Neumann boundary conditions in a ball domain Ω ⊂ R N N ≥ 4 with smooth boundary ∂ Ω . This paper considers in the singular limit ε ⟶ 0 ; the result comes from the finite time blow-up of arbitrary large values of n in the corresponding nonlocal scalar parabolic equation case when N ≥ 4 and χ &gt; 2 N / N − 2 .

Nonlocal fractional elliptic equations and applications

AbstractThe Lp‐coercive properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proved that the fractional elliptic operator generated by this equation is sectorial in Lp space and also is a generator of an analytic semigroup. Moreover, by using the Lp‐separability properties of the given elliptic operator the maximal regularity of the corresponding nonlocal fractional parabolic equation is established.

Linear Galerkin-Legendre spectral scheme for a degenerate nonlinear and nonlocal parabolic equation arising in climatology

A special place in climatology is taken by the so-called conceptual climate models. These relatively simple sets of differential equations can successfully describe single mechanisms of climate. We focus on one family of such models based on the global energy balance. This gives rise to a degenerate nonlocal parabolic nonlinear partial differential equation for the zonally averaged temperature. We construct a fully discrete numerical method that has an optimal spectral accuracy in space and second order in time. Our scheme is based on the Galerkin formulation of the Legendre basis expansion, which is particularly convenient for this setting. By using extrapolation, the numerical scheme is linear even though the original equation is nonlinear. We also test our theoretical results during various numerical simulations that support the aforementioned accuracy of the scheme.

On the dynamics of the roots of polynomials under differentiation

This article is devoted to the study of a nonlinear and nonlocal parabolic equation introduced by Stefan Steinerberger to study the roots of polynomials under differentiation; it also appeared in a work by Dimitri Shlyakhtenko and Terence Tao on free convolution. Rafael Granero-Belinchón obtained a global well-posedness result for initial data small enough in a Wiener space, and recently Alexander Kiselev and Changhui Tan proved a global well-posedness result for any initial data in the Sobolev space Hs(S) with s>3/2. In this paper, we consider the Cauchy problem in the critical space H1/2(S). Two interesting new features, at this level of regularity, are that the equation can be written in the form∂tu+V∂xu+γΛu=0, where V is not bounded and γ is not bounded from below. Therefore, the equation is only weakly parabolic. We prove that nevertheless the Cauchy problem is well posed locally in time and that the solutions are smooth for positive times. Combining this with the results of Kiselev and Tan, this gives a global well-posedness result for any initial data in H1/2(S). Our proof relies on sharp commutators estimates and introduces a strategy to prove a local well-posedness result in a situation where the lifespan depends on the profile of the initial data and not only on its norm.

Solving nonlinear parabolic equations under nonlocal conditions by a nonlocal boundary shape function and splitting-linearizing method

In the article, we solve a nonlinear parabolic type partial differential equation (PDE) subject to non-separated and nonlocal conditions. First, a nonlocal boundary shape function (NLBSF) is derived to satisfy the initial condition and two nonlocal conditions. In the NLBSF, upon letting the free function be the Pascal polynomials the new bases can be created, which automatically fulfill all the conditions specified. The solution is then expanded in terms of these bases. Collocating points inside the space-time domain to satisfy the nonlinear PDE and in conjunction with a novel splitting-linearizing technique, quite accurate solution of the nonlocal and nonlinear parabolic equation can be achieved very fast. The numerical examples are given which confirm the high accuracy and efficiency of the proposed iterative method.

Cahn–Hilliard equations governed by weakly nonlocal conservation laws and weakly nonlocal particle interactions

We consider a doubly nonlocal nonlinear parabolic equation which describes phase segregation of a binary system subject to weak-to-weak interactions [Gal, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018)]. The proposed model reduces to the classical Cahn–Hilliard equation under certain conditions. We establish well-posedness results (based on regular and nonregular mild solutions) along with regularity and long-time results in terms of finite-dimensional attractors. Then we also establish the convergence of (certain) mild solutions to single steady states as time goes to infinity. These results are also supplemented by a handful of (two-dimensional) numerical experiments displaying phase-segregation phenomena with interesting interface morphologies, depending on various choices of the interaction kernels (i.e., Gaussian, logarithmic, Riesz and bimodal potentials). We develop a stable numerical scheme which is able to control the computations under the effect of the double nonlinear convolutions.