We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation $u_t=Δu+|u|^{p-1}u$ which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent $p$ is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution.
We consider the semilinear heat equation $u_t=Δu+u^p$ on ${\mathbb R}^N$. Assuming that $N\ge 3$ and $p$ is greater than the Sobolev critical exponent $(N+2)/(N-2)$, we examine entire solutions (classical solutions defined for all $t\in {\mathbb R}$) and ancient solutions (classical solutions defined on $(-\infty,T)$ for some $T<\infty$). We prove a new Liouville-type theorem saying that if $p$ is greater than the Lepin exponent $p_L:=1+6/(N-10)$ ($p_L=\infty$ if $N\le 10$), then all positive bounded radial entire solutions are steady states. The theorem is not valid without the assumption of …
We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.
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.
We prove a Liouville type result for bounded, entire solutions to a class of variational semilinear elliptic systems, based on the growth of their potential energy over balls with growing radius. Important special cases to which our result applies are the Ginzburg-Landau system and systems that arise in the study of multi-phase transitions.
We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms \[ Δu+u^p\left|\nabla u\right|^q\leq0, \] on weighted graphs, where $(p,q)\in\mathbb{R}^2$. We give a complete classification of $(p,q)$ under which sharp volume growth assumptions are established.
We provide some Liouville theorems for ancient nonnegative solutions of the heat equation on a complete non-compact Riemannian manifold with Ricci curvature bounded from below. We determine growth conditions ensuring triviality of the latters, showing their optimality through examples.
The question of triviality of solutions of the semilinear Ornstein-Uhlenbeck equation, \[ Δw-\frac{1}{2} \langle x,\nabla w\rangle-\fracλ{p-1}w+|w|^{p-1}w=0, \] is considered. It is shown, that if $p>1$ is Sobolev subcritical or critical and $λ\leq 1$, then all bounded entire solutions are constant. Moreover, in the critical case, the same conclusion holds in the subclass of radial solutions provided that $n\geq 4$ and $λ\in \left[\frac{3 n}{2(n-1)},2\right]$.
In this note, we answer affirmatively the question if a warped product of a compact manifold with a line as an ancient solution to the Ricci flow is trivial. We also consider the global behavior of the Type III warping product Ricci flow.
We consider semilinear elliptic second-order partial differential inequalities of the form Lu +|u|q-1u < and = Lv +|v|q-1v (*) in the whole space Rn, where n > and = 2, q > 0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are defined, measurable and locally bounded in Rn, and that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of a capacity associated with the operator L for solutions of (*), which are define…
In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.
We prove that suitable weak solutions of the Navier-Stokes equations exhibit Type I singularities if and only if there exists a non-trivial mild bounded ancient solution satisfying a Type I decay condition. The main novelty is in the reverse direction, which is based on the idea of zooming out on a regular solution to generate a singularity. By similar methods, we prove a Liouville theorem for ancient solutions of the Navier-Stokes equations bounded in $L^3$ along a backward sequence of times.
We obtain a new Liouville comparison principle for entire weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$ u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*) $$ in the half-space ${\mathbb S} = {\mathbb R}^1_+ \times \mathbb R^n$. Here $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i} [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}],$$ where $a_{ij}(t,x)$, $i,j=1,...,n$, are functions defined, measurable and locally bounded in $\mathbb S$, and such that $a_{ij}(t,x)=a_{ji}(t,x)$ …
A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville Theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove in some particular cases the Liouville Theorem. Finally, we give an alternative form of the stress-energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.
In this paper, we consider the Liouville property for ancient solutions of the incompressible Navier-Stokes equations. In 2D and the 3D axially symmetric case without swirl, we prove sharp Liouville theorems for smooth ancient mild solutions: velocity fields $v$ are constants if vorticity fields satisfy certain condition and $v$ are sublinear with respect to spatial variables, and we also give counterexamples when $v$ are linear with respect to spatial variables. The condition which vorticity fields need to satisfy is $\lim\limits_{|x|\rightarrow +\infty}|w(x,t)|=0$ and $\lim\limits_{r\rightar…
We obtain a new Liouville comparison principle for weak solutions $(u,v)$ of semilinear parabolic second-order partial differential inequalities of the form $$u_t -{\mathcal L}u- |u|^{q-1}u\geq v_t -{\mathcal L}v- |v|^{q-1}v (*)$$ in the whole space ${\mathbb E} = {\mathbb R} \times \mathbb R^n$. Here, $n\geq 1$, $q>0$ and $$ {\mathcal L}=\sum\limits_{i,j=1}^n\frac{\partial}{{\partial}x_i}\left [ a_{ij}(t, x) \frac{\partial}{{\partial}x_j}\right],$$ where $a_{ij}(t,x)$, $i,j=1,\dots,n$, are functions that are defined, measurable and locally bounded in $\mathbb E$, and such that $a_{ij}(t,x)=a_…
In this paper, we establish a local gradient estimate for a $p$-Lpalacian equation with a fast growing gradient nonlinearity. With this estimate, we can prove a parabolic Liouville theorem for ancient solutions satisfying some growth restriction near infinity.
We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. The main advantage of our approach is that it allows us to remove a restrictive monotonicity assumption that was imposed in the recent paper. In addition, we can remove a non-degeneracy condition that was assumed in the latter reference. Furtherm…
In this paper, we give a full classification of the nonexistence of positive weak solutions to the semilinear elliptic inequality involving the fractional Hardy potential in punctured and in exterior domains. Our methods are self-contained and new. The main ideas and key ingredients will be discussed in the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4 for exterior domains. We will also explain why all the previous methods and techniques do not apply to our general setting. Let us give here a foretaste of our line of attack: Based on the imbalance between the Hard…
In this paper, We characterize bounded ancient solutions to the time-dependent Stokes system with zero boundary value in various domains, including the half space.
We are concerned with solutions to the nonlinear heat equation $u_t=Δu+|u|^{p-1}u$, $x\in \mathbb{R}^N$, that are defined for all positive and negative time. If the exponent $p$ is greater or equal to the Joseph-Lundgren exponent $p_c$ and $|u|$ stays below some positive radially symmetric steady state, under a mild condition on the behaviour of $u$ as $|x|\to \infty$, we show that $u$ is independent of time. Our method of proof uses Serrin's sweeping principle, based on the strong maximum principle, applied to the linearized equation for $u_t$. Our result covers that of Poláčik and Yanagida […
We derive an adaptation of Li & Yau estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We then apply these estimates to obtain a Harnack inequality and to discuss monotonicity, convexity, decay estimates and triviality of ancient and eternal solutions.
Applying the method of moving planes in integral forms, we establish radial symmetry for positive solutions to a class of semilinear equations involving the fractional Laplacian in the unit ball and obtain Liouville type theorems concerning the non-existence of positive solutions in the half space. The regularity of solutions are also investigated.
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.
Under appropriate assumptions, we show that all bounded entire solutions to a class of semilinear elliptic systems are confined in a convex domain. Moreover, we prove a Liouville type theorem in the case where the domain is strictly convex. Our result represents an extension, under less regularity assumptions, of a recent result of Smyrnelis. We also provide several applications.
We study a class of semilinear elliptic equations with constraints in higher dimension. It is known that several mathematical structures of the problem are closed to those of the Liouville equation in dimension two. In this paper, we establish a classification of entire solutions, the $\sup + \inf$ type inequality and the quantized blowup mechanism.
We establish a Liouville theorem for bounded mild ancient solutions to the axi-symmetric incompressible Navier-Stokes equations on $(-\infty, 0] \times (\mathbb{R}^2 \times \mathbb{T}^1)$. This is a step forward to completely solve the conjecture on $(-\infty, 0] \times \mathbb{R}^3$ which was made in \cite{KNSS} to describe the potential singularity structures of the Cauchy problem.
In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We show that the set of non characteristic points $I_0$ is open and that the blow-up curve is of class $C^{1,μ_0}$ and the phase $θ$ is $C^{μ_0}$ on this set. In order to prove this result, we introduce a Liouville Theorem for that equation.
In this paper, we establish Liouville type results for semilinear subelliptic systems associated with the sub-Laplacian on the Heisenberg group $\mathbb{H}^{n}$ involving two different kinds of general nonlinearities. The main technique of the proof is the method of moving planes combined with some integral inequalities replacing the role of maximum principles. As a special case, we obtain the Liouville theorem for the Lane-Emden system on the Heisenberg group $\mathbb{H}^{n}$, which also appears to be a new result in the literature.
In this paper, we prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R N , as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. We also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities.
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.
We prove a theorem for the growth of the energy of bounded, globally minimizing solutions to a class of semilinear elliptic systems of the form $Δu=\nabla W(u)$, $x\in \mathbb{R}^n$, $n\geq 2$, with $W:\mathbb{R}^m\to \mathbb{R}$, $m\geq 1$, nonnegative and vanishing at exactly one point (at least in the closure of the image of the considered solution $u$). As an application, we can prove a Liouville type theorem under various assumptions.
We study bounded ancient solutions of the Navier-Stokes equations. These are the solutions which are defined for all past time. In two space dimensions we prove that such solutions are either constant or functions of time only, depending on the exact definition of admissible solutions. The general three dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axi-symmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions.
In the paper \cite{KNSS:1}, the authors make the following conjecture: {\it any bounded ancient mild solution of the 3D axially symmetric Navier-Stokes equations is constant.} And it is proved in the case that the solution is swirl free. Our purpose of this paper is to improve their result by allowing that the solution can grow with a power smaller than 1 with respect to the distance to the origin. Also, we will show that such a power is optimal to prove the Liouville type theorem since we can find counterexamples for the Navier-Stokes equations such that the Liouville theorem fails if the sol…
In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in $\mathbb{R}^n$, which can be represented by the Riemann-Liouville fractional integral of order $1-γ$ with $γ\in(0,1)$. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of $γ$, on the blow-up range between the effective case and the non-effective ca…
In this paper, we study the asymptotic behavior as $x_1\to+\infty$ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at $x_1=0$ is given. We prove the uniqueness and characterize the one-dimensional or constant profile of the solutions at infinity. To do so, we use two different approaches. The first one is a pure PDE approach and it is based on the maximum principle, the sliding method and some new Liouville type results for elliptic equations in the half-space or in the whole space~$\mathbb{R}^N$. The second one is based on the theory of dynamical …
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.
Let $0<α,β<2$ be any real number. In this paper, we investigate the following semilinear system involving the fractional Laplacian \begin{equation*} \left\{\begin{array}{lll} (-\lap)^{α/2} u(x)=f(v(x)), & (-\lap)^{β/2} v(x)=g(u(x)), & \qquad x\in\mathbb{R}^n_+, u,v\geq0, & \qquad x\in\mathbb{R}^n\setminus\mathbb{R}^n_+. \end{array}\right. \end{equation*} Applying a direct method of moving planes for the fractional Laplacian, without any decay assumption on the solutions at infinity, we prove Liouville theorems of nonnegative solutions under some natural conditions on $f$ and $g$.
This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solu…
