We establish the uniqueness of positive radial solutions of
$$\begin{cases} Δu
+f(u)=0,\quad x\in A \\ u(x) =0 \quad x\in \partial A
\end{cases} $$ where $A:=A_{a,b}=\{ x\in {\mathbb R}^n : a<|x|<b \}$, $0<a<b\le\infty$.
We assume that the nonlinearity $f\in C[0,\infty)\cap C^1(0,\infty)$ is such that $f(0)=0$ and satisfies some convexity and growth conditions, and either $f(s)>0$ for all $s>0$, or has one zero at $B>0$, is non positive and not identically 0 in $(0,B)$ and it is positive in $(B,\infty)$.
We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A in R^2m ;m >1, invariant by the action of a certain symmetry group can be reduced to a nonhomogenous similar problem in an annulus D in R^(m+1), invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A which concentrate on one or two (m-1) dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration ten…
We consider the uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. We deduce the uniqueness from the argument in the classical paper by Peletier and Serrin, thereby recovering a part of the uniqueness result of Ouyang and Shi.
In this article we find the equivalent conditions to assure the existence and uniqueness of positive solutions to semilinear elliptic equations wih double power nonlinearities. As a bonus, we give a simpler proof of our former result that the uniqueness condition comes from the existence condition.
We consider semilinear elliptic equations with double power nonlineaities. The condition to assure the existence of positive solutions is well-known. In the present paper, we remark that the additional condition to assure uniqueness proposed by Ouyang and Shi is unnecessary.
In this paper we study a semilinear problem for the fractional laplacian that are the counterpart of the Neumann problems in the classical setting. We show uniqueness of minimal energy solutions for small domains.
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 consider uniqueness of positive solutions to semilinear elliptic equations with double power nonlinearities. The condition to assure the existence of positive solutions to these types of equations has long been known. On the other hand for uniqueness, quite technical additional condition is proposed by Ouyang and Shi in 1998. In the present paper we remark that this additional condition is unnecessary.
We quantify the uniqueness of continuation from Cauchy or interior data. Our approach consists in extending the existing results in the linear case. As by product we obtain a new stability estimate in the linear case. We also show the so-called strong uniqueness of continuation and the uniqueness of continuation from a set of positive measure. These results are derived by using a linearization procedure.
Given $Ω$ a bounded open subset of $\mathbb{R}^N$, we consider nonnegative solutions to the singular semilinear elliptic equation $-Δ\,u\,=\,\frac{f}{u^β}$ in $H^1_{loc}(Ω)$, under zero Dirichlet boundary conditions. For $β>0$ and $f\in L^1(Ω)$, we prove that the solution is unique.
An inverse problem to determine a space-dependent factor in a semilinear time-fractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution of the inverse problem is studied. The method uses a positivity principle of the corresponding differential equation that is also proved in the paper.
We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation $(-Δ)^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to $2$. Moreover, we demonstrate the comparison principle and provide a $L^\infty$ estimate for this nonlocal equation under appropriate regularity assumptions.
In this paper we prove symmetry results for classical solutions of nonlinear cooperative elliptic systems in a ball or in annulus in R^N, with N greather or equal than 2. More precisely we prove that solutions having Morse index j smaller than N are foliated Schwarz symmetric if the nonlinearity has a convex derivative and a full coupling condition is satisfied along the solution.
We consider a semilinear elliptic problem in an annulus of R^N, with N>1. Recent results ensure that there exists a sequence p_k of exponents of the nonlinear term at which a nonradial bifurcation from the radial solution occurs. Exploiting the properties of O(N-1)-invariant spherical harmonics, we introduce two suitable cones K^1 and $K^2$ of O(N-1)-invariant functions that allow to separate the branches of bifurcating solutions from the others, getting the unboundedness of these branches.
Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or the…
We establish the uniqueness of the higher radial bound state solutions of $$ Δu +f(u)=0,\quad x\in \RR^n. \leqno(P) $$ We assume that the nonlinearity $f\in C(-\infty,\infty)$ is an odd function satisfying some convexity and growth conditions, and either has one zero at $b>0$, is non positive and not identically 0 in $(0,b)$, and is differentiable and positive $[b,\infty)$, or is positive and differentiable in $[0,\infty)$.
We consider a parabolic equation of the form u_t=Δu +f(u)+h(x,t) in R^N\times (0,\infty), where f in C^1(R) is such that f(0)=0 and f'(0)<0 and h is a suitable function on R^N\times (0,\infty). We show that under certain conditions, each globally defined and nonnegative bounded solution u converges to a single steady state.
In their 1968 paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negative solutions. In particular they showed that if the underlying ODE has non-unique solutions (as characterised via an Osgood-type condition) {\em and} the nonlinearity $f$ satisfies a concavity condition, then the parabolic PDE also inherits the non-uniqueness property. This concavity assumption has remained in place either implicitly or explicitly in all subsequent work in the literature relating to this and other, …
Sharp non-uniqueness of weak solutions to the semilinear wave equation
School of Mathematical Sciences & Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学数学科学学院与当代应用数学上海市重点实验室）；School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
We investigate the initial value problem for a semilinear heat equation with exponential-growth nonlinearity in two space dimension. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space $H^1(\R^2)$. The uniqueness part is non trivial although it follows Brezis-Cazenave's proof \cite{Br} in the case of monomial nonlinearity in dimension $d\geq3$. Next, %Following Caffarelli-Vasseur \cite{cv}, we show that in the defocusing case our solution is bounded, and therefore exists for all time. In the focusing case, we prove that any solution with negative …
A wide variety of articles, starting with the famous paper (Gidas, Ni and Nirenberg in Commun. Math. Phys. 68, 209-243 (1979)) is devoted to the uniqueness question for the semilinear elliptic boundary value problem -Δu=λu+u^p in Ω, u>0 in Ω, u=0 on the boundary of Ω, where λ ranges between 0 and the first Dirichlet Laplacian eigenvalue. So far, this question was settled in the case of Ω being a ball and, for more general domains, in the case λ=0. In (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)), we proposed a computer-assisted approach to this uniqueness question, which indeed pro…
We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to $$ {div}\big(\mathsf A\,\nabla v\big)+\mathsf B\,f(v)=0\,,\quad\lim_{|x|\to+\infty}v(x)=0,\quad x\in\mathbb R^n,$$ $n>2$, where $\mathsf A$ and $\mathsf B$ are two positive, radial, smooth functions defined on $\mathbb R^n\setminus\{0\}$. We assume that the nonlinearity $f\in C(-c,c)$, $0<c\le\infty$ is an odd function satisfying some convexity and growth conditions, and has a zero at $b>0$, is non positive and not identically 0 i…
In this paper we consider a semilinear elliptic equation with a strong singularity at $u=0$, namely $ \displaystyle u\geq 0 \mbox{ in } Ω$, $ \displaystyle - div \,A(x) D u = F(x,u) \mbox{ in} \; Ω$, $u = 0 \mbox{ on} \; \partial Ω$, with $F(x,s)$ a Carathéodory function such that $$ 0\leq F(x,s)\leq \frac{h(x)}{Γ(s)}\,\,\mbox{ a.e. } x\inΩ,\, \forall s>0, $$ with $h$ in some $L^r(Ω)$ and $Γ$ a $C^1([0,+\infty[)$ function such that $Γ(0)=0$ and $Γ'(s)>0$ for every $s>0$. We introduce a notion of solution to this problem in the spirit of the solutions defined by transposition. This definition a…
In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form \begin{align*} \begin{aligned} - Δu &= f(|x|,u,v)\quad\text{in}\R^n, - Δv &= f(|x|,v,u)\quad\text{in}\R^n. \end{aligned} \end{align*} As an application we consider the following nonlinear Schrödinger system \begin{align*} \begin{aligned} - Δu + u &= u^{2q-1} + b u^{q-1}v^q\quad\text{in}\R^n, - Δv + v &= v^{2q-1} + b v^{q-1}u^q \quad\text{in}\R^n. \end{aligned} \end{align*} for $b>0$ and exponents $q$ which satisfy $1<q<\infty$ in case $n\in\{1,2\}$ a…
Let $Ω$ be a $\mathcal C^2$-bounded domain of $\mathbb R^d$, $d=2,3$, and fix $Q=(0,T)\timesΩ$ with $T\in(0,+\infty]$. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation $\partial_t^αu+\mathcal A u=f_b(u)$ in $Q$ where $1<α<2$, $\partial_t^α$ corresponds to the Caputo fractional derivative of order $α$, $\mathcal A$ is an elliptic operator and the nonlinearity $f_b\in \mathcal C^1( \mathbb R)$ satisfies $f_b(0)=0$ and $|f_b'(u)|\leq C|u|^{b-1}$ for some $b>1$. We first provide a definition of local weak solutions of…
We study inverse source problems associated to semilinear elliptic equations of the form \[ Δu(x)+a(x,u)=F(x), \] on a bounded domain $Ω\subset \mathbb{R}^n$, $n\geq 2$. We show that it is possible to use nonlinearity to break the gauge symmetry of the inverse source problem for a class of nonlinearities $a(x,u)$. This is in contrast to inverse source problems for linear equations, which always have a gauge symmetry. The class of nonlinearities include certain polynomials and exponential nonlinearities. For these nonlinearities, we determine both $a(x,u)$ and $F(x)$ uniquely from the associate…
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 equation $-Δu+f(x,u)=0$ in smooth bounded domain $Ω\in\mathbb{R}^N$, $N\geqslant2$, with $f(x,r)>0$ in $Ω\times\mathbb{R}^1_+$ and $f(x,r)=0$ on $\partialΩ$. We find the condition on the order of degeneracy of $f(x,r)$ near $\partialΩ$, which is a criterion of the existence-nonexistence of a very singular solution with a strong point singularity on $\partialΩ$. Moreover, we prove that the mentioned condition is a sufficient condition for the uniqueness of a large solution and conjecture that this condition is also a necessary condition of the uniqueness.
The purpose of this paper is to study the weak solutions of the fractional elliptic problem \begin{equation}\label{000} \begin{array}{lll} (-Δ)^αu+εg(u)=k\frac{\partial^αν}{\partial \vec{n}^α}\quad &{\rm in}\quad\ \ \barΩ,\\[3mm] \phantom{(-Δ)^α+εg(u)} u=0\quad &{\rm in}\quad\ \ \barΩ^c, \end{array} \end{equation} where $k>0$, $ε=1$ or $-1$, $(-Δ)^α$ with $α\in(0,1)$ is the fractional Laplacian defined in the principle value sense, $Ω$ is a bounded $C^2$ open set in $R^N$ with $N\ge 2$, $ν$ is a bounded Radon measure supported in $\partialΩ$ and $\frac{\partial^αν}{\partial \vec{n}^α}$ is defi…
Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the fractional Hardy potential, the growth of the absorption term and the concentration of the measure, in which existence and uniqueness results holds.
We consider an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifold with time-like boundary. The time-dependent coefficients of the nonlinear terms can be recovered in the interior from the knowledge of the Neumann-to-Dirichlet map. Either distorted plane waves or Gaussian beams can be used to derive uniqueness.
We establish a logarithmic stability inequality for the inverse problem of determining the non linear term, appearing in a semilinear BVP, from the corresponding Dirichlet-to-Neumann map (abbreviated to DtN map in the rest of this text). Our result can be seen as a stability inequality for an earlier uniqueness result by Isakov and Sylvester [Commun. Pure Appl. Math. 47 (1994), 1403-1410].
In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.
This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find…
We are mainly concerned with equations of the form $-Lu=f(x,u)+μ$, where $L$ is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, $f$ satisfies the monotonicity condition and mild integrability conditions, and $μ$ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric divergence form operators, with gradient perturbations of some pseudodifferential operators an…
An Inverse Almost Periodic Problem for a Semilinear Strongly Damped Wave Equation
Institute of Mathematics, Humboldt University of Berlin（柏林洪堡大学数学研究所）；Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院应用力学与数学问题研究所）；Department of Computational Mathematics and Programming, Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University（利沃夫国立理工大学应用数学与基础科学研究所计算数学与编程系）；Institute of Mathematics, National Academy of Sciences of Ukraine（乌克兰国家科学院数学研究所）
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.
We investigate the monotonicity method for fractional semilinear elliptic equations with power type nonlinearities. We prove that if-and-only-if monotonicity relations between coefficients and the derivative of the Dirichlet-to-Neumann map hold. Based on the strong monotonicity relations, we study a constructive global uniqueness for coefficients and inclusion detection for the fractional Calderón type inverse problem. Meanwhile, we can also derive the Lipschitz stability with finitely many measurements. The results hold for any $n\geq 1$.
In this article we exploite the uniform decay for damped linear wave equation with Zaremba boundary condition, obtained in a previous work, to treat the same problem in nonlinear context. We need a uniqueness assumption, usual for this type of nonlinear problem. The result is deduced from an observation estimate for nonlinear problem proved by a contradiction argument.
We introduce a probabilistic representation for solutions of quasilinear wave equation with analytic nonlinearities. We use stochastic cascades to prove existence and uniqueness of the solution.
We analyze the semilinear elliptic equation $Δu=ρ(x) f(u)$, $u>0$ in ${\mathbf R}^D$ $(D\ge3)$, with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions $u$ such that $\lim_{|x|\rightarrow +\infty}u(x)=+\infty$. Assuming that $f$ satisfies the Keller-Osserman growth assumption and that $ρ$ decays at infinity in a suitable sense, we prove the existence of entire large solutions. We then discuss the more delicate questions of asymptotic behavior at infinity, uniqueness and symmetry of solutions.
We consider equations of the form $-L_μu +f(u)=0$ in a smooth domain $Ω$, where $L_μ=Δ+ μδ^{-2}$ and $δ(x)$ denotes the distance of the point $x$ to the boundary of the domain. The nonlinear term $f$ is positive, increasing and convex on $(0,\infty)$, satisfies the Keller-Osserman condition and some additional technical assumptions. The conditions are satisfied, in particular, by power and exponential nonlinearities. We discuss the question of existence and uniqueness of large solutions when $μ>0$.
