P. Mastrolia, D. D. Monticelli, F. Punzo
We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.
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An analogy of nonexistence result by Baras and Goldstein (1984), for the heat equation with inverse singular potential, is proved for 2mth-order linear parabolic equations with Hardy-supercritical singular potentials. Extensions to other linear and nonlinear singular PDEs are discussed.
We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $Ω\times(0,+\infty)$, where $Ω$ is a bounded domain. In particular, we investigate how the behavior of the potential near the boundary of the domain and the power nonlinearity affect the nonexistence of solutions. Particular attention is devoted to the special case of the semilinear parabolic problem, for which we show that the critical rate of growth of the potential near the boundary ensuring nonexistence is sharp.
We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted volume growth conditions on geodesic balls.
We reinvestigate nonexistence and existence of global positive solutions to heat equation with a potential term on Riemannian manifolds. Especially, we give a very natural sharp condition only in terms of the volume of geodesic ball to obtain nonexistence results.
Local existence and nonexistence of solutions to the Hardy parabolic equation with general nonlinearity
本文研究了Hardy抛物方程带一般非线性项的局部解存在性与不存在性，通过上解方法给出了初始函数的最优积分性条件。
We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.
Nonexistence Results for a General Class of Parabolic Problems with a Potential on Weighted Graphs
本文研究加权图上一类带正势函数的半线性抛物型方程，通过测试函数方法导出非负非平凡解不存在的显式参数范围，并将已有结果扩展至包含多孔介质方程和快速扩散方程的更广泛背景。
Nonexistence of solutions to parabolic problems with a potential on weighted graphs
本文研究了加权图上带正位势的半线性抛物方程非平凡非负解的不存在性，在距离Laplacian上界和加权时空体积增长条件下证明了全局解不存在，并讨论了假设的最优性以恢复Fujita型临界指数现象。
In this paper we prove a nonexistence result for nonlinear parabolic problems with zero lower order term whose model is $$ \begin{cases} u_{t}- Δ_p u+|u|^{q-1}u=λ& \text{in}\ (0,T)\timesΩ u(0,x)=0 & \text{in}\ Ω,\\ u(t,x)=0 & \text{on}\ (0,T)\times\partialΩ, \end{cases} $$ where $Δ_p ={\rm div }(|\nabla u|^{p-2}\nabla u)$ is the usual $p$-laplace operator, $λ$ is measure concentrated on a set of zero parabolic $r$-capacity ($1<p<r$), and $q$ is large enough.
We are concerned with the nonexistence of sign-changing global weak solutions for a class of semilinear parabolic differential inequalities with convection terms in exterior domains. A weight function of the form $t^α|x|^σ$ is considered in front of the power nonlinearity. Two types of non-homogeneous boundary conditions are investigated: Neumann-type and Dirichlet-type boundary conditions. Using a unified approach, for each case, we establish sufficient criteria for the nonexistence of global weak solutions. When $α=0$, the critical exponent in the sense of Fujita is obtained. This exponent i…
We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-Δ)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce seve…
In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-Δ)^\fracθ{2} u=|x|^{-γ} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)=μ\quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$, $0<θ\le2$, $p>1$, $γ>0$ and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. Using these conditions, we attempt to identify the optimal strength of the singularity of $μ$ for the existence of solutions to this problem.
In this paper, we give a clear cut relation between the the volume growth $V(r)$ and the existence of nonnegative solutions to parabolic semilinear problem \begin{align}\tag{*}\label{*} \left\{ \begin{array}{ll} Δu - \partial_t u + u^p = 0, \\ {u(x,0)= {u_0(x)}}, \end{array} \right. \end{align} on a large class of Riemannian manifolds. We prove that for parameter $p>1$, if \begin{align*} \int^{+\infty} \frac{t}{V(t)^{p-1}} dt = \infty \end{align*} then (\ref{*}) has no nonnegative solution. If \begin{align*} \int^{+\infty} \frac{t}{V(t)^{p-1}} dt < \infty \end{align*} then (\ref{*}) has positi…
In this paper we are concerned with a class of elliptic differential inequalities with a potential both on $\erre^m$ and on Riemannian manifolds. In particular, we investigate the effect of the geometry of the underlying manifold and of the behavior of the potential at infinity on nonexistence of nonnegative solutions.
Nonexistence of Solutions to classes of parabolic inequalities in the Riemannian setting
本文研究了在黎曼流形上抛物型不等式解的不存在性，通过测试函数方法确定了参数范围，揭示了几何结构、非线性幂次与势函数行为之间的相互作用。
In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity $$u_t-kΔu_t-Δu=I^γ_{0+}(|u|^{p})+ω(x),\,\ (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ where $p>1,\,k\geq 0$, $ω(x)\neq0$ and $I^γ_{0+}$ is the left Riemann-Liouville fractional integral of order $γ\in(0,1).$ Based on the test function method, we have proved the blow-up result for the critical case $γ=0,\,p=p_c$ for $N\geq3$, which answers an {\bf open question} posed in \cite{Zhou}, and in particular when $k=0$ it improves the result obtained in \cite{Bandle}. An interesting fact is that in the cas…
Global existence and nonexistence analyses for a magnetic fractional pseudo-parabolic equation
本文研究磁分数阶Orlicz-Sobolev空间中伪抛物方程的初边值问题，利用嵌入定理、位势井理论和Galerkin方法证明了全局解的存在唯一性、衰减估计及有限时间爆破，并探讨了与平稳问题基态解的收敛关系。
In this paper, we are interested to analyze a nonlocal nonlinear parabolic equation with fractional Laplacian. We show that there are no nontrivial global weak solutions using the test function method.
Nonexistence Results for Semilinear Parabolic and Hyperbolic Equations on Metric Graphs
研究度量图上具有正势能的半线性抛物和双曲不等式解的不存在性，通过构造新伪度量和时空测试函数，在加权时空体积增长条件下证明非常弱解必为零。
We show the existence of nodal solutions to perturbed quasilinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds. A nonexistence result is also given.
Nonexistence of solutions to $Δ_pu+Δ_qu+u^s|\nabla u|^t\leq 0$ on geodesically complete noncompact Riemannian manifolds
本文通过测试函数方法，在测地完备非紧黎曼流形上证明了含梯度项的不等式解的Liouville型定理，推广了已有结果，并讨论了高阶项在s<0时的影响。
Existence, Nonexistence, and Symmetry of Positive Solutions for Fractional Laplacian Problems
本文研究含分数阶 Laplace 算子的椭圆与抛物问题，利用山路引理和移动平面法证明次临界正解的存在性及对称性或单调性，并给出超临界与负指数情形的不存在性。
On nonnegative solutions of the parabolic differential inequality with $(p,q)$-Laplace on Riemannian manifolds
本文研究了Riemannian流形上带有$(p,q)$-拉普拉斯算子的抛物型微分不等式中的Liouville型定理，通过测试函数方法，在合适的加权体积增长假设下得出非存在性结果，扩展了抛物型不等式的非存在性结论。
The Parabolic Harnack Inequality on Weighted Riemannian Manifolds
基于Grigor'yan的方法，对一大类抛物型微分算子建立了加权黎曼流形上的抛物型Harnack不等式。
The author extends previous results to general classes of equations under weaker assumptions obtained in 2016 by Bao, Dong and Jiao concerning the study of the regularity of solutions for the first initial-boundary value problem for parabolic Hessian equations on Riemannian manifolds.
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.
In this paper, we are concerned with the first initial boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a priori C^2 estimates is our main part. Based on these estimates, the existence of classical solutions is proved under conditions which are nearly optimal.
We propose a general Noetherian approach to Rellich integral identities. Using this method we obtain a higher order Rellich type identity involving the polyharmonic operator on Riemannian manifolds admitting homothetic transformations. Then we prove a biharmonic Rellich identity in a more general context. We also establish a nonexistence result for semilinear systems involving biharmonic operators.
In this paper, we establish the second order estimates of solutions to the first initial-boundary value problem for general Hessian type fully nonlinear parabolic equations on Riemannian manifolds. The techniques used in this article can work for a wide range of fully nonlinear PDEs under very general conditions.
We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive classical solutions) for the following model problems: the scalar nonlinear heat equation $$ u_t-Δu=u^p \qquad\hbox{in}\ {\mathbb R}^n\times{\mathbb R}, $$ its vector-valued generalization with a $p$-homogeneous nonlinearity and the linear heat equation in ${\mathbb R}^n_+\times{\mathbb R}$ complemented by nonlinear boundary conditions of the form $\partial u/\par…
Galerkin-type methods for strictly parabolic equations on compact Riemannian manifolds
本文利用Galerkin方法的变体证明了紧致黎曼流形上严格抛物型方程（含随机强迫）弱解的存在性，并可用于构造收敛的有限元方法。
On the Nonexistence of Global Solutions for Nonlocal Parabolic Equations with Forcing Terms
本文研究带时空强迫项的非局部半线性抛物方程，揭示非局部非线性与强迫项的相互作用对解适定性的影响，并给出小初值全局存在与不存在的临界指数条件。
In this paper, we derive a priori estimates for the gradient and second order derivatives of solutions to a class of Hessian type fully nonlinear parabolic equations with the first initial-boundary value problem on Riemannian manifolds. These a priori estimates are derived under conditions which are nearly optimal. Especially, there are no geometric restrictions on the boundary of the Riemannian manifolds. And as an application, the existence of smooth solutions to the first initial-boundary value problem even for infinity time is obtained.
In this work we consider viscosity solutions to second order parabolic PDEs $u_{t}+F(t,x,u,du,d^{2}u)=0$ defined on compact Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold $M$ has nonnegative sectional curvature, we get the finest results. If one additionally requires $F$ to depend on $d^{2}u$ in a uniformly continuous manner, the assumptions on curvature can be thrown away.
We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-κ$ for $κ\geq 0$. In the elliptic case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} to nonlinear elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequali…
In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in earlier by the author. This allows for a unified treatment of Sobolev-Slobodeckii and little Hölder spaces. In the flat case we recover classical results for Petrowskii-parabolic Cauchy problems.
In this paper we are concerned with a class of elliptic differential inequalities with a potential in bounded domains both of $\mathbf{R}^m$ and of Riemannian manifolds. In particular, we investigate the effect of the behavior of the potential at the boundary of the domain on nonexistence of nonnegative solutions.
This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.
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。
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.
We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabré's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This…
We study Sobolev estimates for the solutions of parabolic equations acting on a vector bundle, in a complete, compact or non compact, riemannian manifold $M.$ The idea is to introduce geometric weights on $M.$ We get global Sobolev estimates with these weights. As applications, we find and improve "classical results", i.e. results without weights, by use of a Theorem by Hebey and Herzlich. As an example we get Sobolev estimates for the solutions of the heat equation on $p$-forms when the manifold has "weak bounded geometry " of order $1$.
Relaxed uniqueness conditions for the parabolic Schrodinger equation on Riemannian manifolds
本文研究了完备非紧黎曼流形上抛物薛定谔方程 Cauchy 问题的解的唯一性，通过利用相关稳态方程正解的衰减性质，显著松弛了所需的积分可积性条件。
In this article we establish a local parabolic almost monotonicity formula for two phase free boundary problems on Riemannian manifolds, which is an extension of a work of Edquist-Petrosyan.
Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds I: Bakry-Émery curvature bounded below
本文提出一种统一方法，将Bakry-Émery曲率有下界的黎曼流形上半线性抛物型方程的微分Harnack不等式推导转化为求解常微分方程组，并由此得到对数型方程和Yamabe型方程的精确估计及Liouville型定理。
We investigate existence and nonexistence of stationary stable nonconstant solutions, i.e. patterns, of semilinear parabolic problems in bounded domains of Riemannian manifolds satisfying Robin boundary conditions. These problems arise in several models in applications, in particular in Mathematical Biology. We point out the role both of the nonlinearity and of geometric objects such as the Ricci curvature of the manifold, the second fundamental form of the boundary of the domain and its mean curvature. Special attention is devoted to surfaces of revolution and to spherically symmetric manifol…
We present some new ideas to derive {\em a priori} second order estiamtes for a wide class of fully nonlinear parabolic equations. Our methods, which produce new existence results for the initial-boundary value problems in $\bfR^n$, are powerful enough to work in general Riemannian manifolds.
We give necessary and sufficient conditions for the existence of weak solutions of a parabolic problem corresponding to the Kolmogorov operators perturbed by a multipolar inverse square potential with respect to the Gaussian probability measure which is the unique invariant measure for Ornstein-Uhlenbeck type operators. We state the optimality of the constant and, then, the nonexistence of positive exponentially bounded solutions to the parabolic problem.
In this work we establish new equivalences for the concept of $p$-parabolic Riemannian manifolds. We define a concept of comparison principle for elliptic PDE's on exterior domains of a complete Riemannian manifold $M$ and prove that $M$ is $p$-parabolic if and only if this comparison principle holds for the $p$-Laplace equation. We show also that the $p$-parabolicity of $M$ implies the validity of this principle for more general elliptic PDS's and, in some cases, these results can be extended for non $p$-parabolic manifolds or unbounded solutions, provided that some growth of these solutions …
We prove matrix and scalar differential Harnack inequalities for linear parabolic equations on Riemannian and Kähler manifolds.
Optimal Liouville theorems for the Lane-Emden equation on Riemannian manifolds
本文在具有非负Ricci曲率的Riemannian流形上，通过严格证明亚临界Lane-Emden-Fowler方程正解的不存在性，扩展了Gidas-Spruck和Serrin-Zou的工作。
We investigate the stability of time-periodic solutions of semilinear parabolic problems with Neumann boundary conditions. Such problems are posed on compact submanifolds evolving periodically in time. The discussion is based on the principal eigenvalue of periodic parabolic operators. The study is motivated by biological models on the effect of growth and curvature on patterns formation. The Ricci curvature plays an important role.
In this paper, we show the nonexistence results for the Kirchhoff elliptic, parabolic, and hyperbolic type equations on the Heisenberg groups. Also, the pseudo-parabolic and pseudo-hyperbolic equations of the Kirchhoff-type are under consideration. To prove these results we use the test function method. In addition, the analogous results are transferred to the cases of systems. Also, we give some examples of non-local nonlinearities.
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $M\times \mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is studied. We consider also elliptic equations on $M$ with similar conditions at infinity.
The fractional anisotropic Calderón problem for a nonlocal parabolic equation on closed Riemannian manifolds
本文研究闭黎曼流形上非局部抛物型方程的分数阶各向异性Calderón问题，证明通过任意小开圆柱体内的局部源到解映射可在等距意义下确定流形，并借助半群理论分析了非局部抛物型算子的谱性质。
Conformal deformation of a Riemannian metric via an Einstein-Dirac parabolic flow
本文提出一种基于狄拉克-爱因斯坦泛函的约束梯度流，用于形变自旋流形上的黎曼度量，并证明了光滑解的局部适定性，为爱因斯坦-狄拉克问题研究奠定基础。
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 work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying approach to problems with strong degeneracies in the interior or at the boundary.
We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial condition we show non positivity of the limiting energy discrepancy. This in turn allows to prove almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales. Such results will be used in [40] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in space…
We prove global comparison results for the $p$-Laplacian on a $p$-parabolic manifold. These involve both real-valued and vector-valued maps with finite $p$-energy.
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