We are concerned with the following semi-linear polyharmonic equation with integral constraint \begin{align} \left\{\begin{array}{rl} &(-Δ)^pu=u^γ_+ ~~ \mbox{ in }{\mathbb{R}^n},\\ \nonumber &\int_{\mathbb{R}^n}u_+^γdx<+\infty, \end{array}\right. \end{align} where $n>2p$, $p\geq2$ and $p\in\mathbb{Z}$. We obtain for $γ\in(1,\frac{n}{n-2p})$ that any nonconstant solution satisfying certain growth at infinity is radial symmetric about some point in $\mathbb{R}^{n}$ and monotone decreasing in the radial direction. In the case $p=2$, the same results are established for more general exponent $γ\in(1,\frac{n+4}{n-4})$. For the following fractional equation with integral constraint \begin{equation*} \left\{\begin{array}{rl} &(-Δ)^sv=v^γ_+ ~~ \mbox{ in }{\mathbb{R}^n},~~~~\\ &\int_{\mathbb{R}^n}v_+^{\frac{n(γ-1)}{2s}}dx<+\infty,~~~~~ \end{array}\right. \end{equation*} where $s\in(0,1)$, $γ\in (1, \frac{n+2s}{n-2s})$ and $n\geq 2$, we also complete the classification of solutions with certain growth at infinity. In addition, observe that the assumptions of the maximum principle named decay at infinity in \cite{chen} can be weakened slightly. Based on this observation, we classify all positive solutions of two semi-linear fractional equations without integral constraint.
Boggio's formula in balls is known for integer-polyharmonic Dirichlet problems and for fractional Dirichlet problems with fractional parameter less than 1. We give here a consistent formulation for fractional polyharmonic Dirichlet problems such that Boggio's formula in balls yields solutions also for the general fractional case.
We provide a detailed description of the relationships between the fractional Laplacian of order $2s\in(0,n)$ on $\mathbb{R}^n$ and the $\textit{$s$-polyharmonic}$ extension operator to the upper half space $\mathbb{R}^{n+1}_+$.
Let $0<α<2$, $p\geq 1$, $m\in\mathbb{N}_+$. Consider $u$ to be the positive solution of the PDE \begin{equation}\label{abstract PDE} (-Δ)^{\fracα{2}+m} u(x)=u^p(x) \quad\text{in }\mathbb{R}^n. \end{equation} Cao, Dai and Qin( Transactions of the American mathematical society, 2021) showed that, under the condition $u\in\mathcal{L}_α$, the PDE possesses super polyharmonic property $(-Δ)^{k+\fracα{2}}u\geq 0$ for $k=0,1,...,m-1$. In this paper, we show another kind of super polyharmonic property $(-Δ)^k u> 0$ for $k=1,...m$ under different conditions $(-Δ)^mu\in\mathcal{L}_α$ and $(-Δ)^m u\geq 0…
A priori estimates for semilinear higher order elliptic equations usually have to deal with the absence of a maximum principle. This note presents some regularity estimates for the polyharmonic Dirichlet problem that will make a distinction between the influence on the solution of the positive and the negative part of the right-hand side.
We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power $s>0$ of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poisson-type kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if $s$ is a natural number. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of $s$-harmonic functions in terms of Marti…
Let $Ω$ be a bounded open domain on the Euclidean space $\mathbb{R}^{n}$ and $\mathbb{Q}_{+}$ be the set of all positive rational numbers. In 2017, Chen and Zeng investigated the eigenvalues with higher order of the fractional Laplacian $\left.(-Δ)^{s}\right|_Ω$ for $s>0$ and $s \in \mathbb{Q}_{+}$, and they obtained a universal inequality of Yang type(\emph{ Universal inequality and upper bounds of eigenvalues for non-integer poly-Laplacian on a bounded domain, Calculus of Variations and Partial Differential Equations, (2017) \textbf{56}:131}). In the spirit of Chen and Zeng's work, we study …
We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the inhomogeneous term for the solvability of the Cauchy problem.
In this paper, we establish the existence of solutions to fractional semilinear parabolic equations in Besov-Morrey spaces for a large class of initial data including distributions other than Radon measures. We also obtain sufficient conditions for the existence of solutions to viscous Hamilton-Jacobi equations.
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.
In this paper, we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice, $h{\mathbb{Z}}^n.$ Contrary to the Euclidean case that was considered in [2], the discrete fractional Klein-Gordon equation is well-posed in $\ell^2(h{\mathbb{Z}}^n).$ However, we also recover the well-posedness results in the certain Sobolev spaces in the limit of the semiclassical parameter $h\rightarrow 0.$
We study a class of fractional semilinear elliptic equations and formulate the corresponding Calderón problem. We determine the nonlinearity from the exterior partial measurements of the Dirichlet-to-Neumann map by using first order linearization and the Runge approximation property.
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 $\frac{n}{n-2σ} < p < \frac{n+2σ}{n-2σ}$.
The main objective of this paper is analysis of the initial-boundary value problems for the linear and semilinear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type fractional derivative acting in the fractional Sobolev spaces. The boundary conditions are formulated in form of the homogeneous Neumann or Robin conditions. First we deal with the uniqueness and existence of the solutions to these initial-boundary value problems. Then we show a positivity property for the solutions and derive the corresponding compari…
In the whole space $\mathbb R^d$, linear estimates for heat semi-group in Besov spaces are well established, which are estimates of $L^p$-$L^q$ type, maximal regularity, e.t.c. This paper is concerned with such estimates for semi-group generated by the Dirichlet Laplacian of fractional order in terms of the Besov spaces on an arbitrary open set of $\mathbb R^d$.
The semilinear space-time fractional Schrödinger equation is considered. First, we give the explicit form for the fundamental solutions by using the Fox $H$-functions in order to to establish some $L^s$ decay estimates. After that, we give some space-time estimates for the mild solutions from which the local well-posedness is derived on some proper Banach space.
In this paper we show that there exist two different critical exponents for global small data solutions to the semilinear fractional diffusive equation with Caputo fractional derivative in time. The second critical exponent appears if the second data is assumed to be zero. This peculiarity is related to the fact that the order of the equation is fractional. To prove our result, we first derive Lr-Lq linear estimates for the solution to the inhomogeneous linear Cauchy problem and then we apply a contraction argument.
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 $n\geq 1$.
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.
Let $0<α<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{α/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and $\mathbb{R}^n_+$. Applying a direct method of moving planes for the fractional Laplacian, we prove symmetry and nonexistence of positive solutions on $\mathbb{R}^n$ and $\mathbb{R}^n_+$ under mild conditions on $f$.
We prove existence of weak solutions to the obstacle problem for semilinear wave equations (including the fractional case) by using a suitable approximating scheme in the spirit of minimizing movements. This extends the results in [9], where the linear case was treated. In addition, we deduce some compactness properties of concentration sets (e.g. moving interfaces) when dealing with singular limits of certain nonlinear wave equations.
We study the Cauchy problem for the semilinear fractional heat equation $u_{t}=\triangle^{α/2}u+f(u)$ with non-negative initial value $u_{0}\in L^{q}(\mathbb{R}^{n})$ and locally Lipschitz, non-negative source term $f$. For $f$ satisfying the Osgood-type condition $\int_{1}^{\infty}\frac{ds}{f(s)}=\infty$, we show that there exist initial conditions such that the equation has no local solution in $L^{1}_{loc}(\mathbb{R}^{n})$.
We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.
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.
For the following semilinear equation with Hilfer- Hadamard fractional derivative \begin{equation*} \mathcal{D}^{α_1,β}_{a^+} u-Δ\mathcal{D}^{α_2,β}_{a^+} u-Δu =\vert u\vert^p, \qquad t>a>0, \qquad x\inΩ, \end{equation*} where $Ω\subset \mathbb{R}^N$ $(N\geqslant 1)$, $p>1$, $0<α_{2}<α_{1}<1$ and $0<β<1$. $\mathcal{D}^{α_i,β}_{a^+}$ $(i=1,2)$ is the Hilfer- Hadamard fractional derivative of order $α_i$ and of type $β$, we establish the necessary conditions for the existence of global solutions.
In this paper, we investigate the spectral analysis (from the point of view of semi-groups) of discrete, fractional and classical Fokker-Planck equations. Discrete and fractional Fokker-Planck equations converge in some sense to the classical one. As a consequence, we first deal with discrete and classical Fokker-Planck equations in a same framework, proving uniform spectral estimates using a perturbation argument and an enlargement argument. Then, we do a similar analysis for fractional and classical Fokker-Planck equations using an argument of enlargement of the space in which the semigroup …
We study bounded solutions to the fractional equation $(-Δ)^s u + u - |u|^{q-2}u = 0$ in $\mathbb R^n$ for $n\ge2$ and subcritical exponent $q>2$. Applying the variational approach based on concentration arguments and symmetry considerations which was introduced by Lerman, Naryshkin and Nazarov (2020) we construct several types of solutions with various structures (radial, rectangular, triangular, hexagonal, quasi-periodic, breather type, etc.).
In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We will extend well-known results for standard evolution equations such as the blow-up alternative, to the time-fractional evolution equations. We also prove the differentiability with respect to time of the solution when the perturbation is sufficiently smooth. The differentiability enables us to use the maximum principle. The theory on general Banach spaces en…
In this paper, we consider the following fractional logarithmic Schrödinger equation \begin{equation*} \varepsilon^{2s}(-Δ)^s u + V(x)u=u\log |u|^2\ \ \text{in}\ \R^N, \end{equation*} where $\varepsilon>0$, $N\ge 1$, $V(x)\in C(\R^N,[-1,+\infty))$. By introducing an interesting penalized function, we show that the problem has a positive solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$. There is no restriction on decay rates of $V$, especially it can be compactly supported.
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).
New fractional $r$-order seminorms, $TGV^r$, $r\in \mathbb R$, $r\geq 1$, are proposed in the one-dimensional (1D) setting, as a generalization of the integer order $TGV^k$-seminorms, $k\in\mathbb{N}$. The fractional $r$-order $TGV^r$-seminorms are shown to be intermediate between the integer order $TGV^k$-seminorms. A bilevel training scheme is proposed, where under a box constraint a simultaneous optimization with respect to parameters and order of derivation is performed. Existence of solutions to the bilevel training scheme is proved by $Γ$-convergence. Finally, the numerical landscape of …
