Yongqin Liu
In this paper we focus on the global-in-time existence and the pointwise estimates of solutions to the initial value problem for the semilinear dissipative wave equation in multi-dimensions. By using the method of Green function combined with the energy estimates, we obtain the pointwise estimates of the solution.
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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.
In this article we prove optimal pointwise bounds for solutions to the semilinear wave equation with integer powers $p\geq 3$ on Kerr backgrounds with small angular momentum and small initial data.
We generalize the pointwise decay estimates for large data solutions of the defocusing semilinear wave equations which we obtained earlier under restriction to spherical symmetry. Without the symmetry the conformal transformation we use provides only a weak decay. This can, however, in the next step be improved to the optimal decay estimate suggested by the radial case and small data results. This is the first result of that kind.
We obtain sharp pointwise estimates for positive solutions to the equation $-Lu+Vu^q=f$, where $L$ is an elliptic operator in divergence form, $q\in\mathbb{R}\setminus \{0\}$, $f\geq 0$ and $V$ is a function that may change sign, in a domain $Ω$ in $\mathbb{R}^{n}$, or in a weighted Riemannian manifold.
In connection with the weak null condition, Alinhac introduced a sufficient condition for global existence of small amplitude solutions to systems of semilinear wave equations in three space dimensions. We introduce a slightly weaker sufficient condition for the small data global existence, and we investigate the asymptotic pointwise behavior of global solutions for systems satisfying this condition. As an application, the asymptotic behavior of global solutions under the Alinhac condition is also derived.
This article gives an energy decay result for small data solutions to a class of semilinear wave equations in two space dimensions possessing weakly dissipative structure relevant to the Agemi condition.
We study the behavior for $t$ small and positive of $C^{2,1}$ nonnegative solutions $u(x,t)$ and $v(x,t)$ of the system \[0\leq u_t-Δu\leq v^λ\] \[0\leq v_t-Δv\leq u^σ\] in $Ω\times (0,1)$, where $λ$ and $σ$ are nonnegative constants and $Ω$ is an open subset of ${\mathbb R}^n$, $n\ge 1$. We provide optimal conditions on $λ$ and $σ$ such that solutions of this system satisfy pointwise bounds in compact subsets of $Ω$ as $t\to 0^+$. Our approach relies on new pointwise bounds for nonlinear heat potentials which are the parabolic analog of similar bounds for nonlinear Riesz potentials.
We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\leq -Δu\leq f(v)$ $0\leq -Δv\leq g(u)$ in $B_1(0)\backslash\{0\}$ where $f,g:(0,\infty)\to (0,\infty)$ are continuous functions. We provide optimal conditions on $f$ and $g$ at $\infty$ such that solutions of this system satisfy pointwise bounds near the origin. In dimension $n=2$ we show that this property holds if $\log^+ f$ or $\log^+g$ grow at most linearly at infinity. In dimension $n\geq 3$ and under the assumption $f(t)=O(t^λ)$, $g(t)=O(t^σ)$ as $t\to \infty$, ($λ, σ\geq 0$), we obtain a…
We consider the Cauchy problem for systems of semilinear wave equations in two space dimensions. We present a structural condition on the nonlinearity under which the energy decreases to zero as time tends to infinity if the Cauchy data are sufficiently small, smooth and compactly-supported.
The semilinear suspension bridge model with pointwise dissipation
本文通过发展鲁棒半群框架，利用比较原理和紧扰动下本质谱不变性，将耦合系统的稳定性分析简化为辅助系统，从而建立线性系统的指数稳定性，并证明半线性问题的整体适定性和指数衰减，最终证明全局紧吸引子和有限分形维数指数吸引子的存在。
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.
Forced Frequency Locking for Semilinear Dissipative Hyperbolic PDEs
本文研究了小时间周期性扰动对半线性耗散双曲偏微分方程时间周期解的影响，推导出标量方程以解释强制频率锁定现象。
Non-convergence and convergence of bounded solutions to semilinear wave equation with dissipative boundary condition
研究带耗散边界条件的半线性波动方程长时间行为，分析无限平衡点集，给出非收敛解并证明 Łojasiewicz 型内源下解依指数速率收敛。
We derive sharp decay estimates and prove holomorphic extensions for the solutions of a class of semilinear nonlocal elliptic equations with linear part given by a sum of Fourier multipliers with finitely smooth symbols at the origin. Applications concern the decay and smoothness of travelling waves for nonlinear evolution equations in fluid dynamics and plasma physics.
We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison p…
This paper is devoted to the study of semi-stable radial solutions $u\in H^1(B_1)$ of $-Δu=g(u) {in} B_1\setminus \{0\}$, where $g\in C^1(R)$ is a general nonlinearity and $B_1$ is the unit ball of $R^N$. We establish sharp pointwise estimates for such solutions. As an application of these results, we obtain optimal pointwise estimates for the extremal solution and its derivatives (up to order three) of the semilinear elliptic equation $-Δu=λf(u)$, posed in $B_1$, with Dirichlet data $u|_{\partial B_1}=0$, and a continuous, positive, nondecreasing and convex function $f$ on $[0,\infty)$ such t…
Semilinear Diffusion Equations on Infinite Graphs: The Dissipative and Lipschitz Cases
本文研究无限连通加权图上的半线性扩散方程，探讨单调递减和利普希茨连续非线性情况，证明在$\ell^p$空间中初值的解存在性、唯一性和正则性，采用隐式欧拉方案和狄利克雷子图技术，并证明有限时间灭绝和正性。
In this paper, we use Dafermos-Rodnianski's new vector field method to study the asymptotic pointwise decay properties for solutions of energy subcritical defocusing semilinear wave equations in $\mathbb{R}^{1+3}$. We prove that the solution decays as quickly as linear waves for $p>\frac{1+\sqrt{17}}{2}$, covering part of the sub-conformal case, while for the range $2<p\leq \frac{1+\sqrt{17}}{2}$, the solution still decays with rate at least $t^{-\frac{1}{3}}$. As a consequence, the solution scatters in energy space when $p>2.3542$. We also show that the solution is uniformly bounded when $p>\…
We obtain an entire Liouville type theorem to the classical semilinear subcritical elliptic equation on Heisenberg group. A pointwise estimate near the isolated singularity was also proved. The soul of the proofs is an a priori integral estimate, which deduced from a generalized formula of that found by Jerison and Lee.
A note on stochastic semilinear dissipative evolution equations
本文研究了带有加性噪声的半线性随机演化方程的解的存在性和唯一性，其中线性部分为紧收缩半群生成元，非线性部分为递减函数相关的叠加算子。
Interior pointwise $C^{1}$ and $C^{1,1}$ regularity of solutions for general semilinear elliptic equation in nondivergence form
该研究在更弱假设下推导非散度型一般半线性椭圆方程$L^n$-粘性解的内点逐点$C^1$与$C^{1,1}$正则性，推广了非齐次项相关结果，给出无界系数下的正则性判定条件。
A bilinear pointwise tracking optimal control problem for a semilinear elliptic PDE
本文研究了半线性椭圆PDE的双线性点跟踪最优控制问题，推导了最优解的存在性，并证明了局部最优控制的正则性性质。
In this paper we prove some symmetry results for entire solutions to the semilinear equation $-Δu=f(u)$, with $f$ nonincreasing in a right neighbourhood of the origin. We consider solutions decaying only in some directions and we give some sufficient conditions for them to be radially symmetric with respect to those variables, such as periodicity or the pointwise decay of some derivatives.
By introducing new weighted vector fields as multipliers, we derive quantitative pointwise estimates for solutions of defocusing semilinear wave equation in $\mathbb{R}^{1+3}$ with pure power nonlinearity for all $1<p\leq 2$. Consequently, the solution vanishes on the future null infinity and decays in time polynomially for all $\sqrt{2}<p\leq 2$. This improves the uniform boundedness result of the second author when $\frac{3}{2}<p\leq 2$.
In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,\text{Dini}} condition at a boundary point, and the nonhomogeneous term satisfies Dini continuous condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg C^{1,\text{Dini}} domains.
The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuous and the discontinuities are not always crossed transversally. Solutions to the semilinear system are obtained by an approximation argument, relying on Kolmogorov's compactness theorem. Reverting to the original variables, one recovers a solution to the nonlinear wave equation where the total energy…
The aim of this paper is to prove a blow up result of the solution for a semilinear scale invariant damped wave equation under a suitable decay condition on radial initial data. The admissible range for the power of the nonlinear term depends both on the damping coefficient and on the pointwise decay order of the initial data. In addition we give an upper bound estimate for the lifespan of the solution, in terms of the power of the nonlinearity, size and growth of initial data.
We prove the existence of solutions $u$ in $H^1(\mathbb{R}^N,\mathbb{R}^M)$ of the following strongly coupled semilinear system of second order elliptic PDEs on $\mathbb{R}^N$ \[ \mathcal{P}[u] = f(x,u,\nabla u), \quad x\in \mathbb{R}^N, \] whith pointwise constraints. We present the construction of the suitable topoligical degree which allows us to solve the above system on bounded domains. The key step in the proof consists of showing that the sequence of solutions of the truncated system is compact in $H^1$ by the use of the so-called tail estimates.
Global attractors for the Signorini problem with pointwise damping
研究了带有点状阻尼的Signorini问题的全局吸引子存在性，证明了问题的解具有指数衰减性和紧致的全局吸引子。
We prove global pointwise decay estimates for a class of defocusing semilinear wave equations in $n=3$ dimensions restricted to spherical symmetry. The technique is based on a conformal transformation and a suitable choice of the mapping adjusted to the nonlinearity. As a result we obtain a pointwise bound on the solutions for arbitrarily large Cauchy data, provided the solutions exist globally. The decay rates are identical with those for small data and hence seem to be optimal. A generalization beyond the spherical symmetry is suggested.
We prove existence and smoothness of the density of the solution to a nonlinear stochastic heat equation on $L^2(\mathcal{O})$ (evaluated at fixed points in time and space), where $\mathcal{O}$ is an open bounded domain in $\mathbb{R}^d$. The equation is driven by an additive Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be (maximal) monotone, continuously differentiable, and growing not faster than a polynomial. The proof uses tools of the Malliavin calculus combined with methods coming from the theory of maximal mono…
In this paper, we prove some pointwise comparison results between the solutions of some second-order semilinear elliptic equations in a domain $Ω$ of $\R^n$ and the solutions of some radially symmetric equations in the equimeasurable ball $Ω^*$. The coefficients of the symmetrized equations in~$Ω^*$ satisfy similar constraints as the original ones in~$Ω$. We consider both the case of equations with linear growth in the gradient and the case of equations with at most quadratic growth in the gradient. Lastly, we show some improved quantified comparisons when the original domain is not a ball. Th…
We consider semilinear wave equations with small initial data in two space dimensions. For a class of wave equations with cubic nonlinearity, we show the global existence of small amplitude solutions, and give an asymptotic description of the solution as $t \to \infty$ uniformly in $x \in {\mathbb R}^2$. In particular, our result implies the decay of the energy when the nonlinearity is dissipative.
We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation \begin{equation*} u_{t}=Δu + |u|^{p} \end{equation*} on complete Riemannian manifolds of dimension $n \geq 5$ with nonnegative Ricci tensor, when $p$ is smaller than the critical Sobolev exponent $\frac{n+2}{n-2}$.
From the work on the weak-null condition by Lindblad and Rodnianski, it is well-known that `bad' quadratic sourcing terms are allowed to appear in coupled semilinear wave equations in three spatial dimensions, provided that such terms appear as sources for `good' variables and that the good variables feed back into the system via `good' sourcing terms. Motivated by these ideas, in this paper we investigate the small data global existence and pointwise decay of solutions to two systems of coupled wave--Klein-Gordon equations in two spatial dimensions. In particular, we consider critical semilin…
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.
This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equation. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work. The results are based on new weighted vector fields as multipliers applied to regions enclosed by light rays. The key observation for th…
In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some $L^2$ estimates of solutions, which improve those in the previous research [46]. Furthermore, asymptotic profiles of the solution and an approximate relation in a framework of the weighted $L^1$ space are derived. Next, with the aid of the classical energy method and Hardy's inequality, we get singular limit results for an energy and the solution itself. …
Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems
本文研究了半线性双曲系统中时间周期解的霍普夫分岔问题，推导了分岔方向的公式并讨论了耗散性对分岔的影响。
In this paper, we study the asymptotic decay properties for defocusing semilinear wave equations in $\mathbb{R}^{1+2}$ with pure power nonlinearity. By applying new vector fields to null hyperplane, we derive improved time decay of the potential energy, with a consequence that the solution scatters both in the critical Sobolev space and energy space for all $p>1+\sqrt{8}$. Moreover combined with Brézis-Gallouet-Wainger type of logarithmic Sobolev embedding, we show that the solution decays pointwise with sharp rate $t^{-\frac{1}{2}}$ when $p>\frac{11}{3}$ and with rate $t^{ -\frac{p-1}{8}+ε}$ …
In the present paper, we are concerned with the semilinear viscoelastic wave equation subject to a locally distributed dissipative effect of Kelvin-Voigt type, posed on a bounded domain with smooth boundary. We begin with an auxiliary problem and we show that its solution decays exponentially in the weak phase space. The method of proof combines an observability inequality and unique continuation properties. Then, passing to the limit, we recover the original model and prove its global existence as well as the exponential stability.
Second-order Optimality Conditions for Time-Optimal Control Problems Governed by Semilinear Parabolic Equations
本文研究了半线性抛物型方程支配的时间最优控制问题，通过引入局部最优解概念，建立了KKT型的一阶和二阶必要最优性条件及二阶充分条件。
We prove existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the $p$-th moment of solutions depends on the integrability of the initial datum, in the whole range $p \in ]0,\infty[$. Lipschitz continuity of t…
We study the large-time behavior of (weak) solutions to a two-scale reaction-diffusion system coupled with a nonlinear ordinary differential equations modeling the partly dissipative corrosion of concrete (/cement)-based materials with sulfates. We prove that as $t\to\infty$ the solution to the original two-scale system converges to the corresponding two-scale stationary system. To obtain the main result we make use essentially of the theory of evolution equations governed by subdifferential operators of time-dependent convex functions developed combined with a series of two-scale energy-like …
Logarithmic gradient estimate and Universal bounds for semilinear elliptic equations revisited
该研究利用Bernstein方法对满足次临界条件的半线性椭圆方程给出对数梯度与普适有界性估计，建立了曲率有下界流形上基本估计的关联，证明非负曲率下估计的κ-等价性，并得到欧氏空间方程的最优估计与Liouville定理。
We prove that the mild solution to a semilinear stochastic evolution equation on a Hilbert space, driven by either a square integrable martingale or a Poisson random measure, is (jointly) continuous, in a suitable topology, with respect to the initial datum and all coefficients. In particular, if the leading linear operators are maximal (quasi-)monotone and converge in the strong resolvent sense, the drift and diffusion coefficients are uniformly Lipschitz continuous and converge pointwise, and the initial data converge, then the solutions converge.
Global regularity of critical SQG via a pointwise Leibniz commutator formula
作者与Luca Gennaioli提出半拉普拉斯算子莱布尼茨交换子的逐点表示，简化统一相关非线性估计，给出二维临界耗散SQG方程整体正则性的简短证明。
We study pointwise behavior of positive solutions to nonlinear integral equations, and related inequalities, of the type \begin{equation*} u(x) - \int_ΩG(x, y) \, g(u(y)) d σ(y) = h, \end{equation*} where $(Ω, σ)$ is a locally compact measure space, $G(x, y)\colon Ω\times Ω\to [0, +\infty]$ is a kernel, $h \ge 0$ is a measurable function, and $g\colon [0, \infty)\to [0, \infty)$ is a monotone function. This problem is motivated by the semilinear fractional Laplace equation \begin{equation*} (-Δ)^{\fracα{2}} u - g(u) σ= μ\quad \text{in} \, \, Ω, \quad u=0 \, \, \, \text{in} \, \, Ω^c, \end{equa…
The weak null condition on Kerr backgrounds
本文研究了Kerr背景上满足弱零条件的半线性波动方程组，在小初值假设下证明了整体存在性和逐点衰减估计。
Finite Element Error Estimates for Critical Growth Semilinear Problems without Angle Conditions
针对d>=2维空间中满足临界增长条件的半线性椭圆边值问题，通过避免离散极大值原理，直接推导先验误差和逐点估计，消除了对限制性角条件的需求。
The critical exponent for the three-dimensional semilinear wave equation with strong damping
该研究确定了带强阻尼的三维半线性波动方程的临界指数为7/3，解决了2014年的开放问题，通过相关分析得到了不同幂次下解的全局存在与爆破结果。
In this paper we find the critical exponent for the global existence (in time) of small data solutions to the Cauchy problem for the semilinear dissipative evolution equations % \[ u_{tt}+(-Δ)^δu_{tt}+(-Δ)^αu+(-Δ)^θu_t=|u_t|^p, \quad t\geq 0,\,\, x\in\R^n,\] % with $p>1$, $2θ\in [0, α]$ and $δ\in (θ,α]$. We show that, under additional regularity $\left(H^{α+δ}(\R^n)\cap L^{m}(\R^n) \right)\times \left(H^{2δ}(\R^n)\cap L^{m}(\R^n)\right) $ for initial data, with $m\in (1,2]$, the critical exponent is given by $p_c=1+\frac{2mθ}{n}$. The nonexistence of global solutions in the subcritical cases i…
We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate …
This paper treats the solvability of a semilinear reaction-diffusion system, which incorporates transport (diffusion) and reaction effects emerging from two separated spatial scales: $x$ - macro and $y$ - micro. The system's origin connects to the modeling of concrete corrosion in sewer concrete pipes. It consists of three partial differential equations which are mass-balances of concentrations, as well as, one ordinary differential equation tracking the damage-by-corrosion. The system is semilinear, partially dissipative, and coupled via the solid-water interface at the microstructure (pore) …
In this manuscript, a sharp lifespan estimate of solutions to semilinear classical damped wave equation is investigated in one dimensional case, when the sum of initial position and speed is $0$ pointwisely. Especially, an extension of lifespan is shown in this case. Moreover, existence of some global solutions are obtained by a direct computation.
Global existence and pointwise decay for nonlinear waves under the null condition
本文研究了非线性波方程在null条件下的全局存在性与点态衰减，证明了解的衰减率与平坦背景一致，并展示了其精确性。
M-dissipative generalized impedance boundary conditions, discrete spectra, and pointwise multipliers between fractional Sobolev spaces
本文研究了有界Lipschitz域中m-耗散广义阻抗边界条件下声学算子的性质，证明了其紧预解算子与阻抗算子紧性等价，并应用于谱离散性及波动方程。
In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa-Holm equation in time weighted $H^1$ space. First, we derive an equivalent semi-linear system by introducing some new variables, and present the globally conservative solutions of this equation in time weighted $H^1$ space. Second, we show that the peakon solutions are conservative weak solutions in $H^1.$ Finally, given a conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables…
Reconstruction of time-dependent coefficients in a semilinear dynamical Schr{ö}dinger equation
通过Dirichlet-to-Neumann映射，利用一阶和高阶线性化技术，从边界测量中逐点重构半线性薛定谔方程的线性与非线性时间依赖系数。
We propose a theoretical model of a non-local dipersive-dissipative equation which contains as a particular case a large class of non-local PDE's arising from stratified flows. Within this fairly general framework, we study the spatial behavior of solutions proving some sharp pointwise and averaged decay properties as well as some pointwise grow properties.
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