Given a smooth domain $Ω\subset\RR^N$ such that $0 \in \partialΩ$ and given a nonnegative smooth function $ζ$ on $\partialΩ$, we study the behavior near 0 of positive solutions of $-Δu=u^q$ in $Ω$ such that $u = ζ$ on $\partialΩ\setminus\{0\}$. We prove that if $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, then $u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$ and we compute the limit of $\abs{x}^{\frac{2}{q-1}} u(x)$ as $x \to 0$. We also investigate the case $q= \frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.
We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-Δu=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.
We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.
We construct positive weak solutions of a class of semilinear elliptic equation which vanish in suitable trace sense on the boundary of a given smooth bounded N-dimensional domain, but which are singular at prescribed isolated points of the boundary. Similar constructions are carried out for solutions which are singular on any given embedded submanifold of the boundary.
We consider the semilinear wave equation with power nonlinearity in one space dimension. We consider an arbitrary blow-up solution $u(x,t)$, the graph $x\mapsto T(x)$ of its blow-up points and ${\cal S}\subset {\mathbb R}$ the set of all characteristic points. We show that ${\ca S}$ is locally finite.
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.
We study the local propagation of conormal singularities for solutions of semilinear wave equations $\square u = P(y, u)$, where $P(y, u)$ is a polynomial of degree $N \geq 3$ in $u$ with $C^\infty(\mathbb{R}^3_y)$ coefficients. We know from the work of Melrose & Ritter and Bony that if u is conormal to three waves which intersect transversally at point $q$, then after the triple interaction $u(y)$ is a conormal distribution with respect to the three waves and the characteristic cone $Q$ with vertex at $q$. We compute the principal symbol of $u$ at the cone and away from the hypersurfaces. We …
Let $n\geq2$ and $ Ω\subset \mathbb{R}^{n+1}$ be a Lipschitz wedge- like domain . We construct positive weak solutions of the problem $$Δu + u^p = 0 \quad\hbox{in}\, Ω,$$ which vanish in a suitable trace sense on $\partialΩ,$ but which are singular at prescribed "edge" of $Ω$ if $p$ is equal or slightly above a certain exponent $p_0>1$ which depends on $Ω.$ Moreover, in the case which $Ω$ is unbounded, the solutions have fast decay at infinity.
We prove the existence of a solution of (--$Δ$) s u + f (u) = 0 in a smooth bounded domain $Ω$ with a prescribed boundary value $μ$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $μ$ is a Dirac mass, we prove the existence of several critical exponents p.
Our purpose of this paper is to study isolated singular solutions of semilinear Helmholtz equation $$ -Δu-u=Q|u|^{p-1}u \quad{\rm in}\ \ \mathbb{R}^N\setminus\{0\},\ \qquad\lim_{|x|\to0}u(x)=+\infty, $$ where $N\geq 2$, $p>1$ and the potential $Q: \mathbb{R}^N\to (0,+\infty)$ is a Hölder continuous function satisfying extra decaying conditions at infinity. We give the classification of the isolated singularity in the Serrin's subcritical case and then isolated singular solutions is derived with the form $u_k=kΦ+v_k$ via the Schauder fixed point theorem for the integral equation $$v_k=Φ\ast\big…
We study existence and uniqueness of solutions of (E 1) --$Δ$u + $μ$ |x| ^{-2} u + g(u) = $ν$ in $Ω$, u = $λ$ on $\partial$$Ω$, where $Ω$ $\subset$ R N + is a bounded smooth domain such that 0 $\in$ $\partial$$Ω$, $μ$ $\ge$ -- N 2 4 is a constant, g a continuous nondecreasing function satisfying some integral growth condition and $ν$ and $λ$ two Radon measures respectively in $Ω$ and on $\partial$$Ω$. We show that the situation differs considerably according the measure is concentrated at 0 or not. When g is a power we introduce a capacity framework which provides necessary and sufficient cond…
We study the propagation of singularities for semilinear Schrodinger equations with quadratic Hamiltonians, in particular for the semilinear harmonic oscillator. We show that the propagation still occurs along the flow the Hamiltonian flow, but for Sobolev regularities in a certain range and provided the notion of Sobolev-wave front set is conveniently modified. The proof makes use of a weighted version of the paradifferential calculus, adapted to our situation. The results can be regarded as the Schrodinger counterpart of those known for semilinear hyperbolic equations, which hold with the us…
We study the semilinear elliptic equation \begin{equation*} -Δu=u^α|\log u|^β\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset\mathbb{R}^n$ with $n\geq 3$, $\frac{n}{n-2} < α< \frac{n+2}{n-2}$ and $-\infty<β<\infty$. Our main result establishes that nonnegative solution $u\in C^2(B_1\setminus\{0\})$ of the above equation either has a removable singularity at the origin or behaves like \begin{equation*} u(x) = A(1+o(1)) |x|^{-\frac{2}{α-1}} \left(\log \frac{1}{|x|}\right)^{-\fracβ{α-1}}\quad\text{as } x\rightarrow 0, \end{equation*} with \begin{equation*} A=\left[\left(\frac{2…
We study the system $-Δ\mathbf{u}=| \mathbf{u}|^{α-1} \mathbf{u}$ with $1<α\leq\frac{n+2}{n-2}$, where $ \mathbf{u}=(u_1,\dots,u_m)$, $m\geq 1$, is a $C^2$ nonnegative function that develops an isolated singularity in a domain of $\mathbb{R}^n$, $n\geq 3$. Due to the multiplicity of the components of $ \mathbf{u}$, we observe a new Pohozaev invariant other than the usual one in the scalar case, and also a new class of singular solutions provided that the new invariant is nontrivial. Aligned with the classical theory of the scalar equation, we classify the solutions on the whole space as well a…
We consider the semilinear heat equation with Sobolev subcritical power nonlinearity in dimension $N=2$, and $u(x,t)$ a solution which blows up in finite time $T$. Given a non isolated blow-up point $a$, we assume that the Taylor expansion of the solution near $(a,T)$ obeys some degenerate situation labeled by some even integer $m(a)\ge 4$. If we have a sequence $a_n \to a$ as $n\to \infty$, we show after a change of coordinates and the extraction of a subsequence that either $a_{n,1}-a_1 = o((a_{n,2}-a_2)^2)$ or $|a_{n,1}-a_1||a_{n,2}-a_2|^{-β} |\log|a_{n,2}-a_2||^{-α} \to L> 0$ for some $L>0…
The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equation…
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 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 show that singularities form after the interaction of three transversal semilinear conormal waves. Our results hold for space dimensions two and higher, and for arbitrary smooth nonlinearity. The case of two space dimensions in which the nonlinearity is a polynomial was studied by the author and Yiran Wang.
In this paper we classify the isolated singularities of positive solutions to Choquard equation and prove the existence of isolated singular solutions.
We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\le -Δu \le (\frac{1}{|x|^α}* v)^λ$ $0\le -Δv \le (\frac{1}{|x|^β}* u)^σ$ in $B_2(0)\setminus\{0\} \subset R^n$, $n\ge 3$, where $λ,σ\ge 0$ and $α,β\in (0,n)$. A by-product of our methods used to study these solutions will be results on the behavior near the origin of $L^1(B_1(0))$ solutions $f$ and $g$ of the system $0 \le f(x) \le C(|x|^{2-α} + \int_{|y|<1}\frac{ g(y) dy}{|x-y|^{α-2}} )^λ$ $0 \le g(x) \le C(|x|^{2-β} + \int_{|y|<1}\frac{ f(y) dy}{|x-y|^{β-2}} )^σ$ for $0<|x|<1$ where $λ,σ\ge 0…
We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-Δu -\fracμ{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<μ<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1\to 0$, or is determined by the solutions of the linear problem $-Δh -\fracμ{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $μ$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinea…
We study the boundary behaviour of the of (E) $-\Gd u-\myfrac{\xk }{d^2(x)}u+g(u)=0$, where $0<\xk <\frac{1}{4}$ and $g$ is a continuous nonndecreasing function in a bounded convex domain of $\BBR^N$. We first construct the Martin kernel associated to the the linear operator $\CL_{\xk }=-\Gd-\frac{\xk }{d^2(x)}$ and give a general condition for solving equation (E) with any Radon measure $\gm$ for boundary data. When $g(u)=|u|^{q-1}u$ we show the existence of a critical exponent $q_c=q_c(N,\xk )>1$: when $0<q<q_c$ any measure is eligible for solving (E) with $\gm$ for boundary data; if $q\geq …
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.
In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in arXiv:1609.03608 and establish a quantization result for entire solutions of the singular n-Liouville equation.
We consider positive solutions $u$ of the semilinear biharmonic equation $Δ^2 u = |x|^{-\frac{n+4}{2}} g(|x|^\frac{n-4}{2} u)$ in $\mathbb R^n \setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions on the nonlinearity $g$, we show that $|x|^\frac{n-4}{2} u$ is a periodic function of $\ln |x|$ and we classify all such solutions.
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…
We prove the existence of a critical Fujita exponent for a non-homogeneous semilinear heat equation which involves degenerate coefficients. More precisely, in order to give a rather complete theory, we focus on two types of weights $w(x)=|x_1|^a$ or $w(x)=|x|^b$ where $a, b>0$ in a suitable range. The coefficients under consideration admit either a singularity at the origin or a line of singularities. In the latter case, the problem is related to the fractional Laplacian.
We give natural topological conditions on the support of the target measure under which solutions to the optimal transport problem with cost function satisfying the (weak) Ma, Trudinger, and Wang condition cannot have any isolated singular points.
In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving the Hardy-Leray potentials \begin{equation}\label{0} -Δu+\fracμ{|x|^2} u=u^p\quad {\rm in}\quad Ω\setminus\{0\},\qquad u=0\quad{\rm on}\quad \partialΩ. \end{equation} We classify the isolated singularities and obtain the existence, the stability of positive solutions of (\ref{0}). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.
All $(-1)$-homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus north and south poles have been classified in our earlier work as a four dimensional surface with boundary. In this paper, we establish near the no-swirl solution surface existence, non-existence and uniqueness results on $(-1)$-homogeneous axisymmetric solutions with nonzero swirl which are smooth on the unit sphere minus north and south poles.
We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem. The main step is an infinite-dimensional Lyapunov-Schmidt reduction method, that reduces the problem to an (infinite dimensional) Toda type system.
In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-Δ)^σu = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We first use blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When $\sg=1$, this classification result has been proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981).
We study nonnegative classical solutions $u$ of the polyharmonic inequality $-Δ^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these solutions $u$ satisfy a pointwise a priori bound as $x\to 0$.
In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations $(-Δ)^σu=u^{\frac{n+2σ}{n-2σ}}$ with an isolated singularity, where $σ\in (0,1)$. We prove that all the solutions are asymptotically radially symmetric. When $σ=1$, these have been proved in \cite{CGS} by Caffarelli, Gidas and Spruck.
In this paper we report on numerical studies of formation of singularities for the semilinear wave equations with a focusing power nonlinearity $u_{tt} - Δu = u^{p}$ in three space dimensions. We show that for generic large initial data that lead to singularities, the spatial pattern of blowup can be described in terms of linearized perturbations about the fundamental self-similar (homogeneous in space) solution. We consider also non-generic initial data which are fine-tuned to the threshold for blowup and identify critical solutions that separate blowup from dispersal for some values of the e…
In this article we develop a framework for studying parabolic semilinear stochastic evolution equations (SEEs) with singularities in the initial condition and singularities at the initial time of the time-dependent coefficients of the considered SEE. We use this framework to establish existence, uniqueness, and regularity results for mild solutions of parabolic semilinear SEEs with singularities at the initial time. We also provide several counterexample SEEs that illustrate the optimality of our results.
We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3$-space.
The precise asymptotic behaviour of the solutions to the twodimensional curvature equation $Δu=k(z) e^{2 u}$ with $e^{2 u} \in L^1$ for bounded nonnegative curvature functions $-k(z)$ near isolated singularities is obtained.
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of such surfaces in terms of the class of real analytic closed locally convex curves in the 2-sphere with admissible cusp singularities, characterizing when the singularity is actually embedded. In the global setting, we describe the space of peaked spheres in R^3, i.e. compact convex surfaces of constant positive curvature with a finite number of singularities, …
We study the asymptotic behavior of local solutions to the Yamabe equation near an isolated singularity, when the metric is not conformally flat. We prove that, in dimension $6$, any solution is asymptotically close to a Fowler solution, which is an extension of the same result for lower dimensions by F.C. Marques in 2008.
In this paper we study the short time existence problem for the (generalized) Lagrangian mean curvature flow in (almost) Calabi--Yau manifolds when the initial Lagrangian submanifold has isolated conical singularities modelled on stable special Lagrangian cones. Given a Lagrangian submanifold $F_0:L\rightarrow M$ in an almost Calabi--Yau manifold $M$ with isolated conical singularities at $x_1,...,x_n\in M$ modelled on stable special Lagrangian cones $C_1,...,C_n$ in $\mathbb{C}^m$, we show that for a short time there exist one-parameter families of points $x_1(t),... x_n(t)\in M$ and a one pa…
The generalized CR equation $u_{\bar{z}}=au+b\bar{u}+f$ is studied when the coefficients $a$ and $b$ have a finite number of singular points inside the domain. Solutions are constructed via the study of an associated integral operator and the existence of nontrivial solutions of the associated homogeneous equation is established.
We are concerned with the inverse boundary problem of determining anomalies associated with a semilinear elliptic equation of the form $-Δu+a(\mathbf x, u)=0$, where $a(\mathbf x, u)$ is a general nonlinear term that belongs to a Hölder class. It is assumed that the inhomogeneity of $f(\mathbf x, u)$ is contained in a bounded domain $D$ in the sense that outside $D$, $a(\mathbf x, u)=λu$ with $λ\in\mathbb{C}$. We establish novel unique identifiability results in several general scenarios of practical interest. These include determining the support of the inclusion (i.e. $D$) independent of its…
We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a constant driving term and random nonlinearity to model the influence of the obstacle field. For the case of isolated obstacles centered on lattice points and admitting a random strength with exponential tails, we show that the interface propagates with a finite velocity for sufficiently large driving force. The proof consists of a discretization of the evolution equation…
We establish a correspondence on a Riemann surface between hyperbolic metrics with isolated singularities and bounded projective functions whose Schwarzian derivatives have at most double poles and whose monodromies lie in ${\rm PSU}(1,\,1)$. As an application, we construct explicitly a new class of hyperbolic metrics with countably many singularities on the unit disc.
