In this paper, we investigate the asymptotic symmetry and monotonicity of positive solutions to a reaction-diffusion equation in the unit ball, utilizing techniques from elliptic geometry. Firstly, we discuss the properties of solutions in the elliptic space. Then, we establish crucial principles, including the asymptotic narrow region principle.Finally, we employ the method of moving planes to demonstrate the asymptotic symmetry of the solutions.
We study local and global properties of solutions of --$Δ$u = u p + M ||u| q in a domain $Ω$ of R N , in the range min{p, q} > 1 and M $\in$ R. We prove a priori estimates and existence or non-existence of ground states.
We study global properties of positive radial solutions of --$Δ$u = up +M |$\nabla$u|p+1 in RN wherep > 1 and M is a real number. We prove the existence or the non-existence of ground states and of solutions with singularity at 0 according to the values of M and p.
The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue …
By employing the N-barrier method developed in the paper, we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. As an application of this maximum principle, we show under certain conditions, the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new $(1,0,0)$-$(u^{\ast},v^{\ast},0)$ waves are given in terms of the tanh function provided that the parameters satisfy certain conditions.
We study the existence of nonnegative solutions to the Dirichlet problem $\CL^{_{^M}}_{p,q}u:=-Δu+u^p-M|\nabla u|^q=μ$ in a domain $Ω\subset\BBR^N$ where $μ$ is a nonnegative Radon measure, when $p>1$, $q>1$ and $M\geq 0$. We also give conditions under which nonnegative solutions of $\CL^{_{^M}}_{p,q}u=0$ in $Ω\setminus K$ where $K$ is a compact subset of $Ω$ can be extended as a solution of the same equation in $Ω$
We study the equation --div(A(x, u)) = g(x, u, u) + $μ$ where $μ$ is a measure and either g(x, u, u) $\sim$ |u| q 1 u||u| q 2 or g(x, u, u) $\sim$ |u| s 1 u + ||u| s 2. We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure. Finally we connect the potential estimates on the measure with Lipchitz estimates with respect to some Bessel or Riesz capacity.
We study properties of nonnegative functions satisfying (E)$\;-Δu+u^p-M|\nabla u|^q=0$ is a domain of $\mathbb{R}^N$ when $p>1$, $M>0$ and $1<q<p$. We concentrate our analysis on the solutions of (E) with an isolated singularity, or in an exterior domain, or in the whole space. The existence of such solutions and their behaviours depend strongly on the values of the exponents $p$ and $q$ and in particular according to the sign of $q-\frac{2p}{p+1}$, and when $q=\frac{2p}{p+1}$, also on the value of the parameter $M$ which becomes a key element. The description of the different behaviours is ma…
We establish quantitative asymptotic behavior of positive solutions of a family of nonlinear elliptic equations on the half cylinder near the end. This unifies the study of isolated singularities of some semilinear elliptic equations, such as the Yamabe equation, Hardy-Hénon equation etc.
We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with logarithmic Kirchhoff function. We establish the precise asymptotic formulas for the solution $u_λ(x)$ as $λ\to \infty$. Here, $λ> 0$ is the bifurcation parameter.
In this paper we investigate the behavior of a family of steady state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a $ε$-neighborhood of a portion $Γ$ of the boundary. We assume that this $ε$-neighborhood shrinks to $Γ$ as the small parameter $ε$ goes to zero. Also, we suppose the upper boundary of this $ε$-strip presents a highly oscillatory behavior. Our main goal here is to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. I…
We consider elliptic equations in planar domains with mixed boundary conditions of Dirichlet-Neumann type. Sharp asymptotic expansions of the solutions and unique continuation properties from the Dirichlet-Neumann junction are proved.
This paper completes and partially improves some of the results of [arXiv:0809.5002] about the asymptotic behavior of solutions of linear and nonlinear elliptic equations with singular coefficients via an Almgren type monotonicity formula
The paper is devoted to the study of asymptotic behavior of solutions for nonlocal elliptic problems in weighted spaces. We deal with the most difficult case where the support of nonlocal terms intersects with the boundary of a plane bounded domain. In this situation, a general form of the asymptotics is investigated and coefficients in the asymptotics are calculated.
We report on some recent existence and uniqueness results for elliptic equations subject to Dirichlet boundary condition and involving a singular nonlinearity. We take into account the following types of problems: (i) singular problems with sublinear nonlinearity and two parameters; (ii) combined effects of asymptotically linear and singular nonlinearities in bifurcation problems; (iii) bifurcation for a class of singular elliptic problems with subquadratic convection term. In some concrete situations we also establish the asymptotic behaviour of the solution around the bifurcation point. Our …
We investigate asymptotic behavior of solutions for nonlocal elliptic boundary value problems in plane angles and in ${\mathbb R}^2\backslash\{0\}$. Such problems arise as model ones when studying asymptotics of solutions for nonlocal elliptic problems in bounded domains. We obtain explicit formulas for the asymptotic coefficients in terms of eigenvectors and associated vectors of both adjoint nonlocal operators (acting in spaces of distributions) and formally adjoint (with respect to the Green formula) nonlocal problems.
We study asymptotic behavior of sub-solutions to non-uniformly elliptic equations with nonstandard growth. In particular, Harnack type inequalities are proved. Our approach gives new results for the cases with (p,q) nonlinearity and generalized Orlicz growth.
This paper studies the solutions of a reaction--diffusion system with nonlinearities that generalise the Lengyel--Epstein and FitzHugh--Nagumo nonlinearities. Sufficient conditions are derived for the global asymptotic stability of the system's solutions and confirmed through numerical Examples.
We consider a reaction-diffusion system which may serve as a model for a ferment catalytic reaction in chemistry. The model consists of a system of reaction diffusion equations with unbounded time dependent coefficients and different polynomial reaction terms. An exponential decay of the globally bounded solutions is proved. The key tool of the proofs are properties of analytic semigroups and some inequalities.
This paper is concerned with investigating the asymptotic behavior of the gradients of solutions to a class of elliptic systems with general boundary data, especially covering the Lamé systems, in a narrow region. The novelty of this paper lies in finding the correction term to improve the previous gradient estimates to be a precise asymptotic formula. Through this justification, we capture all singular terms and thus give a fairly sharp characterization in terms of the singularities of the gradients.
Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations.
In this paper we obtain existence results for the positive solution of a singular elliptic boundary value problem. To prove the main results we use comparison arguments and the method of sub-super solutions combined with a procedure which truncates the singularity.
We analyze the asymptotic behavior of eigenvalues and eigenfunctions of an elliptic operator with mixed boundary conditions on cylindrical domains when the length of the cylinder goes to infinity. We identify the correct limiting problem and show in particular, that in general the limiting behavior is very different from the one for the Dirichlet boundary conditions.
We study operators on a singular manifold, here of conical or edge type, and develop a new general approach of representing asymptotics of solutions to elliptic equations close to the singularities. The idea is to construct so-called asymptotic parametrices with flat left -over terms. Our structures are motivated by models of particle physics with singular potentials that contribute embedded singularities in $\R^N$ of higher order, according to the number of particles.
In this paper, we derive new asymptotic expansions for the solutions of higher order elliptic equations in the presence of small inclusions. As a byproduct, we derive a topological derivative based algorithm for the reconstruction of piecewise smooth functions. This algorithm can be used for edge detection in imaging, topological optimization, and for inverse problems, such as Quantitative Photoacoustic Tomography, for which we demonstrate the effectiveness of our asymptotic expansion method numerically.
In this paper, we prove the exact asymptotic behavior of singular positive solutions of fractional semi-linear equations $$(-Δ)^σu = u^p~~~~~~~~in ~~ B_1\backslash \{0\}$$ with an isolated singularity, where $σ\in (0, 1)$ and $\frac{n}{n-2σ} < p < \frac{n+2σ}{n-2σ}$.
We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and $k$-Hessian operators.
We will present some rigidity results for solutions to semilinear elliptic equations of the form $\Deltau = W'(u)$, where W is a quite general potential with a local minimum and a local maximum. We are particularly interested in Liouvlle-type theorems and symmetry results, which generalise some known facts about the Cahn-Hilliard equation.
In this work we establish local $C^{2,α}$ regularity estimates for flat solutions to non-convex fully nonlinear elliptic equations provided the coefficients and the source function are of class $C^{0,α}$. For problems with merely continuous data, we prove that flat solutions are locally $C^{1,\text{Log-Lip}}$.
Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \[-Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\,x\in \mathbb{R}^{N},\] where $1<p<N, 0\leqμ<\left({(N-p)}/{p}\right)^{p}$ and $Q\in L^{\infty}(\mathbb{R}^{N})$.
Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), & & x\in\R^{N}, \end{eqnarray*} where $1<p<N$, $0\leqμ<\left((N-p)/p\right)^{p}$, $m>0$ and $f$ is a continuous function.
This paper devoted to study of fractional elliptic equations driven a multiplicative noise. By combining the eigenfunction expansion method for symmetry elliptic operators, the variation of constant formula for strong solutions to scalar stochastic fractional differential equations, Ito's formula and establishing a new weighted norm associated with a Lyapunov-Perron operator defined from this representation of solutions, we show the asymptotic behaviour of solutions to these systems in mean square sense. As a consequence, we also prove existence, uniqueness and the convergence rate of their so…
In this paper we consider the existence of positive solutions for a singular elliptic problem involving an asymtotically linear nonlinearity and depending on one positive parameter. Using variational methods, together with comparison techniques, we show the existence, uniqueness, non-existence and regularity of the solutions. We also obtain a bifurcation-type result.
By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of {\it asymptotic average solutions} making pointwise solvable every Poisson equation $\mathcal{L} u(x)=-f(x)$ with continuous data $f$, where $\mathcal{L}$ is a hypoelliptic linear partial differential operator with positive semidefinite characteristic form.
We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type monotonicity formulae and provides a classification of all possible homogeneity degrees of limiting entire profiles. As a consequence, we establish a strong unique continuation principle from boundary points.
We analyze the asymptotic behavior of the eigenvalues of nonlinear elliptic problems under Dirichlet boundary conditions and mixed (Dirichlet, Neumann) boundary conditions on domains becoming unbounded. We make intensive use of Picone identity to overcome nonlinearity complications. Altogether the use of Picone identity makes the proof easier with respect to the known proof in the linear case. Surprisingly the asymptotic behavior under mixed boundary conditions critically differs from the case of pure Dirichlet boundary conditions for some class of problems.
Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves move. This value depends on the solution concept being considered. We analyze an extensive class of Fisher-type reaction-diffusion equations with flows in divergence form. We work with regular flows, which may not meet the standard elliptical conditions, but without other types of singularities. We show that the range of speeds at which classic traveling waves move is an interval unbounded to the right. Contrary to classic examples, the infimum may no…
We study the large-time behavior of the free boundary position capturing the one-dimensional motion of the carbonation reaction front in concrete-based materials. We extend here our rigorous justification of the $\sqrt{t}$-behavior of reaction penetration depths by including non-linear effects due to deviations from the classical Henry's law and time-dependent Dirichlet data.
Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), & & x\in\R^{N}, \end{eqnarray*} where $1<p<N,0\leqμ<\left((N-p)/p\right)^{p}$, $m>0$ and $f$ is a nonlinear function.
In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (Ann. Inst. H. Poincare Anal. Non Lineaire, 1991) and Rey (J. Funct. Anal., 1990).
We construct solutions of nonlinear reaction-diffusion equations with nonlinear boundary conditions in spaces where the problem is supercritical and show the nonlinear balance required between the nonlinear terms in order to obtain a dissipative system. Assuming this balance, the dynamics of the solutions is the same that take place in subcritical spaces.
In this paper we study a semilinear elliptic problem on a bounded domain in $\R^2$ with large exponent in the nonlinear term. We consider positive solutions obtained by minimizing suitable functionals. We prove some asymtotic estimates which enable us to associate a "limit problem" to the initial one. Usong these estimates we prove some quantitative properties of the solution, namely characterization of level sets and nondegeneracy.
In this paper we report the asymptotic behaviors of viscosity solutions of the following degenerate elliptic equations \begin{equation*}\label{main-Eq} Lu=x_n^{2α}\sum_{i,j=1}^{n-1}a_{ij}(x)D_{ij}u(x) +2x_n^α\sum_{i=1}^{n-1}a_{in}(x)D_{in}u(x)+D_{nn}u(x)=0 \end{equation*} outside a bounded domain of the upper half space for $α>0$ with proper coefficients. Meanwhile, the Hölder estimates up to the boundary will be obtained using rescaling methods.
This work is devoted to study of a class of elliptic singular perturbed systems and their singular limit to a phase segregating system. We prove existence and uniqueness and study the asymptotic behaviour with convergence to a limiting problem as the interaction rate tends to infinity. The limiting problem is a free boundary problem such that at each point in the domain at least one of the components is zero which implies simultaneously all components can not coexist. We present a novel method, which provides an explicit solution of limiting problem for special choice of parameters. Moreover, …
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 …
This paper concerns with existence, uniqueness and asymptotic behavior of the solutions for a nonlocal coupled system of reaction-diffusion. We prove the existence and uniqueness of weak solutions by the Faedo-Galerkin method and exponential decay of solutions by the classic energy method. We improve the results obtained by Chipot-Lovato and Menezes for coupled systems. A numerical scheme is presented.
In this paper, we are concerned with the existence and asymptotic behavior of minimizers for a minimization problem related to some quasilinear elliptic equations. Firstly, we proved that there exist minimizers when the exponent $q$ equals to the critical case $q^*=2+\frac{4}{N}$, which is different from that of \cite{cjs}. Then, we proved that all minimizers are compact as $q$ tends to the critical case $q^*$ when $a<a^*$ is fixed. Moreover, we studied the concentration behavior of minimizers as the exponent $q$ tends to the critical case $q^*$ for any fixed $a>a^*$.
We establish a lower bound for the number of sign changing solutions with precisely two nodal domains to the singularly perturbed nonlinear elliptic equation -ε^{2}Δ_{g}u+u=|u|^{p-2}u on an n-dimensional Riemannian manifold M, p in (2,2n/(n-2)), in terms of the cup-length of the configuration space of M. We give a precise description of the asymptotic profile of these solutions as ε goes to 0. We also provide new estimates for the cup-length of the configuration space of M.
We characterize an asymptotic mean value formula in the viscosity sense for the double phase elliptic equation $$ -{\rm div}(\lvert \nabla u \rvert^{p-2}\nabla u+ a(x)\lvert\nabla u \rvert^{q-2}\nabla u)=0 $$ and the normalized double phase parabolic equation $$ u_t=\lvert\nabla u \rvert ^{2-p}{\rm div}(\lvert \nabla u \rvert^{p-2}\nabla u+ a(x,t)\lvert\nabla u \rvert^{q-2}\nabla u), \quad 1<p\leq q<\infty. $$ This is the first mean value result for such kind of nonuniformly elliptic and parabolic equations. In addition, the results obtained can also be applied to the $p(x)$-Laplace equations …
A semilinear singularly perturbed reaction-diffusion equation with Dirichlet boundary conditions is considered in a convex unbounded sector. The singular perturbation parameter is arbitrarily small, and the "reduced equation" may have multiple solutions. A formal asymptotic expansion for a possible solution is constructed that involves boundary and corner layer functions. For this asymptotic expansion, we establish certain inequalities that are used in a subsequent paper to construct sharp sub- and super-solutions and then establish the existence of a solution to a similar nonlinear elliptic p…
We study the asymptotic behavior of the nonlinear parabolic flows $u_{t}=F(D^2 u^m)$ when $t\ra \infty$ for $m\geq 1$, and the geometric properties for solutions of the following elliptic nonlinear eigenvalue problems: F(D^2 \vp) &+ μ\vp^{p}=0, \quad \vp>0\quad\text{in $Ω$} \vp&=0\quad\text{on $\pΩ$} posed in a (strictly) convex and smooth domain $Ω\subset\re^n$ for $0< p \leq 1,$ where $F(\cdot)$ is uniformly elliptic, positively homogeneous of order one and concave. We establish that $\log (\vp)$ is concave in the case $p=1$ and that the function $\vp^{\frac{1-p}{2}}$ is concave for $0<p<1.$
