Marek Fila, Johannes Lankeit
We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all $t>0$. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as $t\to\infty$.
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A threshold result was proved in this paper for semilinear parabolic system with pure power type nonlinearities
We characterize the asymptotic speed of propagation of almost planar solutions to a semilinear viscous parabolic equation, with periodic nonlinearity.
Monotonicity results for semilinear parabolic equations on metric graphs
该研究证明了局部有限连通度量图上半线性抛物问题的单调性结果，并探讨了其在正则度量树上的应用。
We obtain pointwise estimates for solutions of semilinear parabolic equations with a potential on connected domains both of $\mathbb R^n$ and of general Riemannian manifolds.
We obtain an upper bound on the initial blow-up of nonnegative solutions of second order semilinear parabolic inequalities when a superlinear exponent in the inequalities is not too large.
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。
Stability of backward-in-time semilinear coupled parabolic systems
本文研究有界区域中半线性耦合抛物型系统的反向问题，利用带权函数$e^{λt}$的修正Carleman估计方法，证明了线性与半线性强耦合系统的条件稳定性估计。
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.
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 establish the existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces for a large class of initial data including distributions other than Radon measures. We also obtain sufficient conditions for the existence of solutions to viscous Hamilton-Jacobi equations.
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 are concerned with global existence for semilinear parabolic equations on Riemannian manifolds with negative sectional curvatures. A particular attention is paid to the class of initial conditions which ensure existence of global solutions. Indeed, we show that such a class is crucially related to the curvature bounds.
We prove an almoste representation formula for positive solutions of semilinear heat equations with power-type absorption the initial trace of which is the indicatrix function of a compact set. The representations involves a Wiener test via Bessel capacities.
It is known that there is a class of semilinear parabolic equations for which interior gradient blow-up (in finite time) occurs for some solutions. We construct a continuation of such solutions after gradient blow-up. This continuation is global in time and we give an example when it never becomes a classical solution again.
In this paper, we study initial boundary value problem of semilinear parabolic equations and prove that any positive solution of its steady-state problem is an initial datum threshold for the existence and nonexistence of global solution to the above mentioned parabolic problem
In this paper, we consider an inverse problem to determine a semilinear term of a parabolic equation from a single boundary measurement of Neumann type. For this problem, a reconstruction algorithm is established by the spectral representation for the fundamental solution of heat equation with homogeneous Neumann boundary condition and the Whitney extension theorem.
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 establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.
Extinction and propagation phenomena for semilinear parabolic equations on metric trees
该研究针对正则度量树T上带KPP型强迫项的半线性热方程柯西-诺依曼问题，分析了解的传播、灭绝特性及传播渐近速度，丰富了度量树结构下半线性抛物方程的动力学研究。
This paper is concerned with the asymptotic behavior of the solution to the semilinear parabolic equation with dynamical boundary condition. Our main goal is to prove the convergence of a global solution to an equilibrium as time goes to infinity by means of a suitable Łojasiewicz--Simon type inequality.
We study the uniqueness of singular radial (forward and backward) self-similar positive solutions of the equation $u_t-Δu = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ where $p\geq(n+2)/(n-2)_+$.
We exhibit a singularly perturbed parabolic problems for which the asymptotic behavior can be described by an one-dimensional ordinary differential equation. We estimate the continuity of attractors in the Hausdorff metric by rate of convergence of resolvent operator.
A semilinear parabolic problem of second order with an unknown time-convolution kernel is considered. The missing kernel is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is addressed. We design a numerical algorithm based on Rothe's method, derive a priori estimates and prove convergence of iterates towards the exact solution.
For a given semilinear parabolic equation with polynomial nonlinearity, many solutions blow up in finite time. For a certain large class of these equations, we show that some of the solutions which do not blow up actually tend to equilibria. The characterizing property of such solutions is a finite energy constraint, which comes about from the fact that this class of equations can be written as the $L^2$ gradient of a certain functional.
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。
Eternal solutions of parabolic equations (those which are defined for all time) are typically rather rare. For example, the heat equation has exactly one eternal solution -- the trivial solution. While solutions to the heat equation exist for all forward time, they cannot be extended backwards in time. Nonlinearities exasperate the situation somewhat, in that solutions may form singularities in both backward and forward time. However, semilinear parabolic equations can also support nontrivial eternal solutions. This article shows how nontrivial eternal solutions can be constructed for a semili…
Rate of convergence of attractors for semilinear singularly perturbed problems: scalar parabolic equations with localized large diffusion
本文研究半线性抛物方程在局部大扩散下的吸引子收敛速率，证明吸引子连续变化并给出收敛速率，需精确描述局部大扩散。
The asymptotic behavior of some semilinear parabolic PDEs is analyzed by means of a "mean value" property. This property allows us to determine, by means of appropriate {\em{a priori}} estimates, some exponential decay results for suitable global solutions. We also apply the method to investigate a well-known finite time blow-up result. An application is given to a one-dimensional semilinear parabolic PDE with boundary degeneracy. Our results shed further light onto the problem of determining initial data for which the corresponding solution is guaranteed to exponentially decay to zero or blow…
Global solutions for semilinear parabolic evolution problems with Hölder continuous nonlinearities
该研究证明了带至多线性增长Hölder连续非线性项的半线性抛物演化方程对广泛初值存在全局强解，并将结果应用于丛林火灾波前传播模型与反应扩散系统。
In this paper, we present output feedback boundary stabilization for a class of semilinear parabolic PDEs with a boundary measurement and an actuation located at the same place. The method uses backstepping transformations, where the state and error systems are proved to be locally exponentially stable in the $\mathbb{H}^4$ norm. The stability of the transformed systems are obtained by constructing a strict Lyapunov function. A numerical example using the FitzHugh-Nagumo equation shows the proposed control law stabilizes the system into its equilibrium solution.
We present an existence result for a partial differential inclusion with linear parabolic principal part and relaxed one-sided Lipschitz multivalued nonlinearity in the framework of Gelfand triples. Our study uses discretizations of the differential inclusion by a Galerkin scheme, which is compatible with a conforming finite element method, and we analyze convergence properties of the discrete solution sets.
If $p>1+2/n$ then the equation $u_t-Δu = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ possesses both positive global solutions and positive solutions which blow up in finite time. We study the large time behavior of radial positive solutions lying on the borderline between global existence and blow-up.
In this article, we consider a semilinear pseudo parabolic heat equation with the nonlinearity which is the product of logarithmic and polynomial functions. Here we prove the global existence of solution to the problem for arbitrary dimension $n \geq 1$ and power index $p>1$. Asymptotic behaviour of the solution has been addressed at different energy levels. Moreover, we prove that the global solution indeed decays with an exponential rate. Finally, sufficient conditions are provided under which blow up of solutions take place.
We investigate the large-time behavior of the sign-changing solution of the inhomogeneous semilinear heat equation with a forcing term depending of time and space. we identify the critical exponent for this problem, which separates the nonexistence/existence of global-in-time solutions, and show the discontinuity of this critical exponent at the point which the inhomogeneous term becomes independent of time variable.
Elliptic regularization of some semilinear parabolic free boundary problems
该研究首次将椭圆正则化方法用于抛物型障碍问题，通过一致能量估计、取弱极限两步证明半线性抛物型自由边界问题强解存在性，还建立了相关非退化估计与最优正则性结果。
We introduce several notions of generalised principal eigenvalue for a linear elliptic operator on a general unbounded domain, under boundary condition of the oblique derivative type. We employ these notions in the stability analysis of semilinear problems. Some of the properties we derive are new even in the Dirichlet or in the whole space cases. As an application, we show the validity of the hair-trigger effect for the Fisher-KPP equation on general, uniformly smooth domains.
In this work, we study the stable determination of four space-dependent coefficients appearing in a coupled semilinear parabolic system with variable diffusion matrices subject to dynamic boundary conditions which couple intern-boundary phenomena. We prove a Lipschitz stability result for interior and boundary potentials by means of only one observation component, localized in any arbitrary open subset of the physical domain. The proof mainly relies on some new Carleman estimates for dynamic boundary conditions of surface diffusion type.
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 a general hyperbolic model of chemotaxis in the multidimensional case. For this system we show the global existence of smooth solutions to the Cauchy problem and we determine their asymptotic behavior. Since this model does not enter in the classical framework of dissipative problems, we analyze it combining the features of the hyperbolic and the parabolic parts and using detailed decay estimates of the Green function.
The blow-up rate estimate for the solution to a semilinear parabolic equation $u_t=Δu+V(x) |u|^{p-1}u$ in $Ω\times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data $u(x,0)=M\vf (x)$ as $M$ goes to infinity, which have been found in \cite{cer}, are improved under some reasonable and weaker conditions compared with \cite{cer}.
Let $G=(V,E)$ be a locally finite connected weighted graph, and $Ω$ be an unbounded subset of $V$. Using Rothe's method, we study the existence of solutions for the semilinear heat equation $\partial_tu+|u|^{p-1}\cdot u=Δu~(p\ge1)$ and the parabolic variational inequality \begin{eqnarray*} \int_{Ω^\circ} \partial_tu\cdot(v-u)\,dμ\ge \int_{Ω^\circ}(Δu+f)\cdot(v-u)\,dμ\qquad\mbox{for any }v\in \mathcal{H}, \end{eqnarray*} where $\mathcal{H}=\{u\in W^{1,2}(V):u=0\mbox{ on }V\backslashΩ^\circ\}$.
We consider a system of seminlinear parabolic variational inequalities with time-dependent convex obstacles. We prove the existence and uniqueness of its solution. We also provide a stochastic representation of the solution and show that it can be approximated by the penalization method. Our proofs are based upon probabilistic methods from the theory of Markov processes and the theory of backward stochastic differential equations.
Existence of solutions for time fractional semilinear parabolic equations in Besov--Morrey spaces
本文研究时间分数阶半线性热方程的柯西问题，在 Besov--Morrey 空间中给出解存在的充分条件，涵盖已有结果并适用于广泛初值。
In this paper we study a simple non-local semilinear parabolic equation with Neumann boundary condition. We give local existence result and prove global existence for small initial data. A natural non increasing in time energy is associated to this equation. We prove that the solution blows up at finite time $T$ if and only if its energy is negative at some time before $T$. The proof of this result is based on a Gamma-convergence technique.
We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast.
Convergence of semilinear parabolic flows with general initial data
本文研究了半线性抛物型方程在一般初始数据下的收敛性，证明流收敛到唯一基态，并加强了之前的相关收敛结果。
Attractor Continuity for Semilinear Parabolic Equations on Thin Domains with Degenerating Outward Peaks
研究半线性抛物方程在退化向外尖峰薄域上吸引子的连续性及收敛速度。
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 …
Local controllability of free boundary three-dimensional semilinear radial parabolic equations
针对具有Stefan边界条件和径向对称数据的自由边界三维半线性热方程，通过简化为一维形式并适配Carleman不等式，结合Schauder不动点定理证明其局部零可控性，为该问题在一维以上空间维度提供首个可控性结果。
Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1Δu+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2Δv+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\ (u(\cdot,0),v(\cdot,0))=(μ,ν), & $x\in{\bf R}^N,$ } \] where $D_1$, $D_2>0$, $0<p\le q$ with $pq>1$ and $(μ,ν)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in ${\bf R}^N$. In this paper we study sufficient conditions on the initial data for the solvability of problem~(P) and clarify optimal singularities of the initial functions for the solvability.
In the first part of the paper we prove various results on regularity of Feynman-Kac functionals of Hunt processes associated with time dependent semi-Dirichlet forms. In the second part we study the Cauchy problem for semilinear parabolic equations with measure data involving operators associated with time-dependent forms. Model examples are non-symmetric divergence form operators and fractional laplacians with possibly variable exponents. We first introduce a definition of a solution resembling Stampacchia's definition in the sense of duality and then, using the results of the first part, we…
In this paper, we study inverse boundary problems associated with semilinear parabolic systems in several scenarios where both the nonlinearities and the initial data can be unknown. We establish several simultaneous recovery results showing that the passive or active boundary Dirichlet-to-Neumann operators can uniquely recover both of the unknowns, even stably in a certain case. It turns out that the nonlinearities play a critical role in deriving these recovery results. If the nonlinear term belongs to a general $C^1$ class but fulfilling a certain growth condition, the recovery results are …
We study the first vanishing time for solutions of the Cauchy-Dirichlet problem to the semilinear $2m$-order ($m \geq 1$) parabolic equation $u_t+Lu+a(x) |u|^{q-1}u=0$, $0<q<1$ with $a(x) \geq 0$ bounded in the bounded domain $Ω\subset \R^N$. We prove that if $N>2m$ and $\displaystyle \int_0^1 s^{-1} \text{meas} \{x \in Ω: |a(x)| \leq s \}^\frac{2m}{N} ds < + \infty$, then the solution $u$ vanishes in a finite time. When $N=2m$, the condition becomes $\displaystyle \int_0^1 s^{-1} (\text{meas} \{x \in Ω: |a(x)| \leq s \}) (-\ln \text{meas} \{x \in Ω: |a(x)| \leq s \}) ds < + \infty$.
In this paper, we study the boundary value problem of the classical semilinear parabolic equations $$ u_t-Δu=|u|^{p-1}u, \ \ in \ \ Ω\times (0,T) $$ and $u=0$ on the boundary $\partialΩ\times [0,T)$ and $u=ϕ$ at $t=0$, where $Ω\subset R^n$ is a compact $C^1$ domain, $1<p\leq p_S$ is a fixed constant, and $ϕ\in C^2_0(Ω)$ is a given smooth function. Introducing new idea, we show that there are two sets $\tilde{W}$ and $\tilde{Z}$ such that for $ϕ\in W$, there is a global positive solution $u(t)\in \tilde{W}$ with $h^1$ omega limit $\{0\}$ and for $ϕ\in \tilde{Z}$, the solution blows up at finite…
We consider the Cauchy problem of fifth order dispersive equations on the torus. We assume that the initial data is sufficiently smooth and the nonlinear term is a polynomial depending on $\partial_x^3 u, \partial_x^2 u, \partial_x u$ and $u$. We prove that the local well-posedness holds on $[-T,T]$ when the nonlinear term satisfies a condition and otherwise, the local well-posedness holds with a smoothing effect only on either $[0,T]$ or $[-T,0]$ and nonexistence result holds on the other time interval, which means that the nonlinear term can not be treated as a perturbation of the linear par…
This paper shows how abstract resolvent estimates imply local smoothing for solutions to the Schrödinger equation. If the resolvent estimate has a loss when compared to the optimal, non-trapping estimate, there is a corresponding loss in regularity in the local smoothing estimate. As an application, we apply well-known techniques to obtain well-posedness results for the semi-linear Schrödinger equation.
We give a global bilateral estimate on the maximal solution $\bar u_F$ of $ \prt_tu-Δu+u^q=0$ in $\BBR^N\times (0,\infty)$, $q>1$, $N\geq 1$, which vanishes at $t=0 $ on the complement of a closed subset $F\subset \BBR^N$. This estimate is expressed by a Wiener test involving the Bessel capacity $C_{2/q,q'}$. We deduce from this estimate that $\bar {u}_F$ is $σ$-moderate in Dynkin's sense.
A monotonicity formula for a semilinear fractional parabolic equation
针对全分数阶半线性热方程，研究人员通过高维抛物-椭圆变换建立了对应的延拓问题的单调性公式，该公式是局部方程Giga-Kohn单调性公式的分数阶类似物，其证明方法或可推广至其他非线性非局部场景。
We study the Cauchy problem for the semilinear nonautonomous parabolic equation $u_t=\mathcal{A}(t)u+ψ(t,u)$ in $[s,τ]\times {{\mathbb R}^d}$, $τ> s $, in the spaces $C_b([s, τ]\times{{\mathbb R}^d})$ and in $L^p((s, τ)\times{{\mathbb R}^d}, ν)$. Here $ν$ is a Borel measure defined via a tight evolution system of measures for the evolution operator $G(t,s)$ associated to the family of time depending second order uniformly elliptic operators $\mathcal{A}(t)$. Sufficient conditions for existence in the large and stability of the null solution are also given in both $C_b$ and $L^p$ contexts. The …
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