Christoph Walker
Existence of strong solutions to a nonlocal semilinear heat equation is shown. The main feature of the equation is that the nonlocal term depends on the unknown on the whole time interval of existence, the latter being given a priori. The proof relies on Schauder's fixed point theorem and semigroup theory.
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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
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.
We consider a possibly anisotropic integro-differential semilinear equation, run by a nondecreasing and nontrivial nonlinearity. We prove that if the solution grows at infinity less than the order of the operator, then it must be constant.
In this paper, we study a class of semilinear nonlocal elliptic equations posed on settings without compact Sobolev embedding. More precisely, we prove the existence of infinitely many solutions to the fractional Brezis-Nirenberg problems on bounded domain.
In this paper we consider a semilinear parabolic equation with nonlinear and nonlocal boundary condition and nonnegative initial datum. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.
We deal with a class of semilinear nonlocal differential equations in Hilbert spaces which is a general model for some anomalous diffusion equations. By using the theory of integral equations with completely positive kernel together with local estimates, some existence, regularity and stability results are established. An application to nonlocal partial differential equations is shown to demonstrate our abstract results.
In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.
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。
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。
We consider the following critical semilinear nonlocal equation involving the fractional Laplacian $$ (-Δ)^su=K(|x|)|u|^{2^*_s-2}u,\ \ \hbox{in}\ \ \mathbb{R}^N, $$ where $K(|x|)$ is a positive radial function, $N>2+2s$, $0<s<1$, and $2^*_s=\frac{2N}{N-2s}$. Under some asymptotic assumptions on $K(x)$ at an extreme point, we show that this problem has infinitely many non-radial positive or sign-changing solutions.
We consider entire solutions to $L u= f(u)$ in $\RR^2$, where $L$ is a general nonlocal operator with kernel $K(y)$. Under certain natural assumtions on the operator $L$, we show that any stable solution is a 1D solution. In particular, our result applies to any solution $u$ which is monotone in one direction. Compared to other proofs of the De Giorgi type results on nonlocal equations, our method is the first successfull attempt to use the Liouville theorem approach to get flatness of the level sets.
Mixed local and nonlocal semilinear elliptic equation with strongly singular and critical Choquard nonlinearity
本文研究混合局部与非局部椭圆方程，结合奇异与临界非线性，通过变分方法证明正解的存在性与多重性。
Controllability of a Semilinear System of Parabolic Equations with Nonlocal Terms
本文研究了具有非局部项的半线性抛物型方程组的可控性问题，通过结合不动点定理和线性化动力学估计，建立了在固定时间内的局部零可控性，并拓展了线性系统的可控性范围。
On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity
该研究利用变分法与逼近方法，考察兼具局部与非局部 Laplace 算子的奇异半线性椭圆方程，证明参数依赖情形下弱解存在及扰动情形下多解存在。
A semilinear version of parabolic-elliptic Keller-Segel system with the \emph{critical} nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a logistic term with a parameter $r$, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large init…
Well-posedness and inverse problems for semilinear nonlocal wave equations
本文研究半线性非局部波动方程，建立适定性，并利用外部Dirichlet-to-Neumann映射和被动测量恢复非线性项及初始数据，结果适用于任意维数。
We consider a linear nonlocal heat equation in a bounded domain $Ω\subset\mathbb{R}^d$ with Dirichlet boundary conditions. The non-locality is given by the presence of an integral kernel. We analyze the problem of controllability when the control acts on an open subset of the domain. It is by now known that the system is null-controllable when the kernel is time-independent and analytic or, in the one-dimensional case, in separated variables. In this paper, we relax this assumption and we extend the result to a more general class of kernels. Moreover, we get explicit estimates on the cost of n…
We investigate the decay for $|x|\rightarrow \infty$ of weak Sobolev type solutions of semilinear nonlocal equations $Pu=F(u)$. We consider the case when $P=p(D)$ is an elliptic Fourier multiplier with polyhomogeneous symbol $p(ξ)$ and derive sharp algebraic decay estimates in terms of weighted Sobolev norms. In particular, we state a precise relation between the singularity of the symbol at the origin and the rate of decay of the corresponding solutions. Our basic example is the celebrated Benjamin-Ono equation {equation} \label{BO}(|D|+c)u=u^2, \qquad c>0,{equation} for internal solitary wav…
In this paper, we consider an elliptic operator obtained as the superposition of a classical second-order differential operator and a nonlocal operator of fractional type. Though the methods that we develop are quite general, for concreteness we focus on the case in which the operator takes the form $-Δ+(-Δ)^s$, with $s\in(0,1)$. We focus here on symmetry properties of the solutions and we prove a radial symmetry result, based on the moving plane method, and a one-dimensional symmetry result, related to a classical conjecture by G.W. Gibbons.
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。
We consider entire solutions to $\mathcal{L}u=f(u)$ in $\mathbb R^2$, where $\mathcal L$ is a nonlocal operator with translation invariant, even and compactly supported kernel $K$. Under different assumptions on the operator $\mathcal L$, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.
Existence Theory for a class of semilinear mixed local and nonlocal equations involving variable singularities and singular measures
本文针对含可变奇异性和奇异测度的半线性局部-非局部混合方程，通过处理可变奇异指数与测度值数据（含新的奇异测度数据），建立了其弱解的存在性。
Symmetry, existence and regularity results for a class of mixed local-nonlocal semilinear singular elliptic problem via variational characterization
本文利用变分刻画和分解性质，结合比较原理与移动平面方法，研究一类混合局部-非局部奇异椭圆问题弱解的对称性、存在性、不存在性与正则性，并用非光滑临界点理论分析跳跃问题。
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
Regularity and existence for semilinear mixed local-nonlocal equations with variable singularities and measure data
本文证明了一类具有可变奇异指数和测度数据的混合局部-非局部奇异问题弱解的存在性与正则性，其中源项可同时为测度，该现象在常数奇异指数情形下也是新的。
Interior regularity of mixed local-nonlocal parabolic semilinear equations
本文证明混合局部-非局部扩散算子半线性抛物方程经典解的存在唯一性，并建立其空间与时间 Hölder 连续性及空间 $C^{2,\alpha}_{loc}$ 内部正则性。
Semilinear Elliptic Systems with Nonlocal Homogeneous Critical Nonlinearities
本文研究推广Brezis-Nirenberg问题的半线性椭圆方程组，确定了其非平凡解的存在条件与仅存在平凡解的条件，丰富了非局部临界非线性椭圆问题的理论研究。
Blow-up of solutions to semilinear parabolic equations driven by mixed local-nonlocal operators with large initial data
本文研究了由混合局部-非局部算子驱动的半线性抛物型方程的非负解在有限时间内爆破的问题，通过改进Kaplan方法证明了当初始数据足够大时解会在有限时间内爆破，尤其在f(u)=u^p情况下对p>1的所有情况均成立。
Global in-time behavior for the semilinear nonlocal heat exchanger system with different diffusion orders
研究具有分数阶拉普拉斯算子的非局部反应扩散系统的全局时间渐近行为，在超临界情形下证明全局存在性，并在一般L^m框架下给出全局解的渐近轮廓，同时确定次临界和临界情形下局部解生命跨度的下界估计。
In this paper we establish the sharp asymptotic decay of positive solutions of the Yamabe type equation $\frlap u=u^{\frac{Q+2s}{Q-2s}}$ in a homogeneous Lie group, where $\frlap$ represents a suitable pseudodifferential operator modelled on a class of nonlocal operators arising in conformal CR geometry.
Blow-up criteria for the semilinear parabolic equations driven by mixed local-nonlocal operators
本文研究混合局部–非局部算子半线性热方程在一般时间相关非线性项下的爆破问题，给出全局解不存在的充要条件，并恢复临界 Fujita 指数。
Homogenization of an optimal control problem for nonlocal semilinear elasticity with soft inclusions
本文研究了非局部半线性弹性体中优化控制问题的均质化，通过Γ收敛方法证明了微观最优控制的极限为极限问题的最优控制。
We consider the the pointwise estimates and the blow-up rate estimates for the zero Dirchilet problem of the semilinear heat equation with a gradient term.
This paper is concerned with finite blow-up solutions of a one dimensional complex-valued semilinear heat equation. We provide locations and the number of blow-up points from the viewpoint of zeros of the solution.
We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the inhomogeneous term for the solvability of the Cauchy problem.
We generalize the notion of renormalized solution to semilinear elliptic and parabolic equations involving operator associated with general (possibly nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We show that under mild integrability assumption on the data a quasi-continuous function $u$ is a renormalized solution to an elliptic (or parabolic) equation in the sense of our definition iff $u$ is its probabilistic solution, i.e. $u$ can be represented by a suitable nonlinear Feynman-Kac formula. This implies in particular that for a broad class of local and nonlocal …
Two families of sign-changing solutions of higher-order semilinear parabolic equations are studied.
Existence of solutions to a fractional semilinear heat equation in uniformly local weak Zygmund type spaces
本文通过引入均匀局部弱Zygmund型空间，建立了临界分数阶半线性热方程解存在性的最优充分条件。
We study the Cauchy problem for a semilinear heat equation with initial data non-rarefied at $\infty$. Our interest lies in the discussion of the effect of the non-rarefied factors on the life span of solutions, and some sharp estimates on the life span is established.
Fujita-type results for the semilinear heat equations driven by mixed local-nonlocal operators
本文研究了由混合局部-非局部算子驱动的半线性热方程的Fujita型临界指数，改进了现有研究并补充了相关工作。
Existence of global solutions to initial value problems for a discrete analogue of a d-dimensional semilinear heat equation is investigated. We prove that a parameter αin the partial difference equation plays exactly the same role as the parameter of nonlinearity does in the semilinear heat equation. That is, we prove non-existence of a non-trivial global solution for 0 < α\le 2/d, and, for α> 2/d, existence of non-trivial global solutions for sufficiently small initial data.
Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In this paper, we study the Dirichlet problem for superlinear equation (E) ${\mathbb L} u = u^p +λμ$ in a bounded domain $Ω$ with homogeneous boundary or exterior Dirichlet condition, where $p>1$ and $λ>0$. The operator ${\mathbb L}$ belongs to a class of nonlocal operators including typical types of fractional Laplacians and the datum $μ$ is taken in the optimal …
Existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term
本文研究了分数半线性热方程中带有奇异非齐次项的解的存在性，通过建立分数热半群的衰减估计和实插值法，获得了精确的积分估计和存在条件。
Recovering Initial States in Semilinear Parabolic Problems from Time-Averages
本文研究半线性抛物问题的非局部初值适定性，并据此由任意时段的小时间平均值恢复初始状态。
We construct a solution for a class of strongly perturbed semilinear heat equations which blows up in finite time with a prescribed blow-up profile. The construction relies on the reduction of the problem to a finite dimensional one and the use of index theory to conclude.
Large solutions to semilinear equations for subordinate Laplacians in $C^{1,1}$ bounded open sets
本文针对由 Laplacian 从属化得到的一类非局部算子，在 $C^{1,1}$ 有界开集中证明半线性问题大解的存在性，并给出以从属子和源项 $f$ 表述的非局部 Keller-Osserman 条件。
We consider semilinear parabolic equations with nonlinear boundary conditions. We give conditions which guarantee global existence of solutions as well as blow-up in finite time of all solutions with nontrivial initial data. The results depend on the behavior of variable coefficients as $t \to \infty.$
We study the initial trace problem for positive solutions of semilinear heat equations with strong absorption. We show that in general this initial trace is an outer regular Borel measure. We emphasize in particular the case where $u$ satisfies (E) $\partial_t u-Δu+t^α|u|^{q-1}u=0$, with $q>1$ and $α>-1$ and prove that in the subcritical case $1
We consider a semilinear parabolic equation with flux at the boundary governed by a nonlinear memory. We give some conditions for this problem which guarantee global existence of solutions as well as blow up in finite time of all nontrivial solutions. The results depend on the behavior of variable coefficients as $t \to \infty.$
Initial traces of solutions to a semilinear heat equation under the Dirichlet boundary condition
该研究探讨Dirichlet边界条件下半线性热方程非负解初始迹的定性性质，针对Cauchy–Dirichlet问题得到非负解存在的严格充要条件，确定可解非负初值的最优奇性。
We consider the semilinear heat equation with a superlinear power nonlinearity in the Sobolev subcritical range. We construct a solution which blows up in finite time only at the origin, with a completely new blow-up profile, which is cross-shaped. Our method is general and extends to the construction of other solutions blowing up only at the origin, with a large variety of blow-up profiles, degenerate or not.
We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a "Newtonian" nonlocal term.
We study the Cauchy problem for the semilinear fractional heat equation $u_{t}=\triangle^{α/2}u+f(u)$ with non-negative initial value $u_{0}\in L^{q}(\mathbb{R}^{n})$ and locally Lipschitz, non-negative source term $f$. For $f$ satisfying the Osgood-type condition $\int_{1}^{\infty}\frac{ds}{f(s)}=\infty$, we show that there exist initial conditions such that the equation has no local solution in $L^{1}_{loc}(\mathbb{R}^{n})$.
We give a simple proof of a lower bound for the Dirichlet heat kernel in terms of the Gaussian heat kernel. Using this we establish a non-existence result for semilinear heat equations with zero Dirichlet boundary conditions and initial data in $L^q(Ω)$ when the source term $f$ is non-decreasing and $\limsup_{s\to\infty}s^{-γ}f(s)=\infty$ for some $γ>q(1+2/n)$. This allows us to construct a locally Lipschitz $f$ satisfying the Osgood condition $\int_{1}^{\infty}1/f(s)\ \,\d s =\infty$, which ensures global existence for bounded initial data, such that for every $q$ with $1\le q<\infty$ there i…
Examples of optimal Hölder regularity in semilinear equations involving the fractional Laplacian
本文研究一维分数阶 Laplace 半线性方程，构造并指出由非局部性与非线性共同导致的解的最优 Hölder 正则性退化现象。
We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.
We consider in this work some class of strongly perturbed for the semilinear heat equation with Sobolev sub-critical power nonlinearity. We first derive a Lyapunov functional in similarity variables and then use it to derive the blow-up rate. We also classify all possible asymptotic behaviors of the solution when it approaches to singularity. Finally, we describe precisely the blow-up profiles corresponding to these behaviors.
In this paper, we establish blow-up rates for higher-order semilinear parabolic equations with nonlocal in time nonlinearity with no positive assumption on the solution. We also give Liouville-type theorem for higher-order semilinear parabolic equation with infinite memory nonlinear term which plays the main tools to prove our blow-up rate result. Finally, we study the well-posedness of mild solutions.
The purpose in this paper is to determine the global behavior of solutions to the initial-boundary value problems for energy-subcritical and critical semilinear heat equations by initial data with lower energy than the mountain pass level in energy spaces associated with self-adjoint operators satisfying Gaussian upper bounds. Our self-adjoint operators include the Dirichlet Laplacian on an open set, Robin Laplacian on an exterior domain, and Schrödinger operators, etc.
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the corresponding Cauchy--Dirichlet problem. Our necessary conditions and sufficient conditions are sharp and enable us to find optimal singularities of initial data for the solvability of the Cauchy--Dirichlet problem.
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