The present paper is concerned with the Cauchy-Dirichlet problem for fractional (and non-fractional) nonlinear diffusion equations posed in bounded domains. Main results consist of well-posedness in an energy class with no sign restriction and convergence of such (possibly sign-changing) energy solutions to asymptotic profiles after a proper rescaling. They will be proved in a variational scheme only, without any use of semigroup theories nor classical quasilinear parabolic theories. Proofs are self-contained and performed in a totally unified fashion for both fractional and non-fractional cases as well as for both porous medium and fast diffusion cases.
We prove two maximal regularity results in spaces of continuous and Hölder continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^α$. This derivative is intended in the sense of Caputo and $α$ is taken in $(0, 2)$. In case $α= 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.
We present two new proofs of the exchange theorem for the Laplace transformation of vector-valued distributions. We then derive an explicit solution to the Dirichlet problem of the polyharmonic operator in a half-space. Finally, we obtain explicit solutions to Cauchy-Dirichlet problems of iterated wave- and Klein-Gordon-operators in half-spaces.
To study the existence and uniqueness of solutions to Cauchy-type problems for fractional q-difference equations with the bi-ordinal Hilfer fractional q-derivative which is an extension of the Hilfer fractional q-derivative. An approach is based on the equivalence of the nonlinear Cauchy-type problem with a nonlinear Volterra q-integral equation of the second kind. Applying an analog of Banach fixed point theorem we prove the uniqueness and the existence of the solution. Moreover, we present an explicit solution to the q-analog of the Cauchy problem for the linear case.
The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation is analyzed. With the focus on non-homogeneous boundary data, two approaches are offered: one is based on the theory of hyperbolic equations, while the other one uses the theory of operator semigroups. This is a mixed hyperbolic problem with a characteristic spatial boundary. Hence, the regularity results exhibit some deficiencies when compared with the non-characteristic case.
In this paper, we consider the Cauchy-type problem (1.1) involving Hilfer-Hadamard-type fractional derivative for a nonlinear fractional differential equation. We prove an equivalence between the Cauchy-type problem (1.1) and Volterra integral equation(VIE), existence, and uniqueness. We present a slight generalization for the Gronwall inequality which was used in studying the continuous dependence of a solution for the Cauchy-type problem (1.2).
We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We obtain decay estimates for large time in Lebesgue spaces.
In this paper, we consider the Cauchy problem for the fractional Camassa-Holm equation which models the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. Using Kato's semigroup approach for quasilinear evolution equations, we prove that the Cauchy problem is locally well-posed for data in $H^{s}({\Bbb R})$, $s>{\frac{5}{2}}$.
In this paper, with the help of previously constructed self-similar solutions, a solution of the Cauchy problem for an equation of even order with a fractional Riemann-Liouville derivative of order $1<α<2$ is obtained.
In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana-Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana-Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.
This paper investigates the Cauchy problem of the time-space fractional Keller-Segel-Navier- Stokes model, which can describe both memory effect and Lévy process of the system. The local existence and global existence in Lebesgue space are obtained by means of Banach fixed point theorem and Banach implicit function theorem, respectively. In addition, the regularities of local and global mild solutions are improved in fractional homogeneous Sobolev spaces. Furthermore, some properties of mild solutions including mass conservation, decay estimates, stability and self-similarity are established.
In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated to a family equations of ZK-KP-type \[ \begin{cases} u_{t}=u_{xxx}-\mathscr{H}D_{x}^αu_{yy}+uu_{x}, \cr u(0)=ψ\in Z \end{cases} \] in anisotropic Sobolev spaces, where $1\le α\le 1$, $\mathscr{H}$ is the Hilbert transform and $D_{x}^α$ is the fractional derivative, both with respect to $x$.
In the present work, we consider the Cauchy problem for the time fractional diffusion equation involving the general Caputo-type differential operator proposed by Kochubei. First, the existence, the positivity and the long time behavior of solutions of the equation without source term are established by using the Fourier analysis technique. Then, based on the representation of the solution of the inhomogenous linear ordinary differential equation with the general operator, the similar problems for the diffusion equation with source term are studied.
In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet-Neumann boundary data when dealing with the Spectral Fractional Laplacian.
We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic Hölder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator can be different with respect to different groups of space variables. The time derivative can be either fractional Caputo - Jrbashyan derivative or usual derivative. Under some necessary conditions on the order of the time derivative we show that the operator of the whole problem is an isomorphism of appropriate anisotropic Hölder spaces. Under some another con…
We study Cauchy problem for the Klein-Gordon (HNLKG), wave (HNLW) and Schrödinger (HNLS) equations with cubic convolution (Hartree type) nonlinearity. Some global well-posedness and scattering are obtained for the (HNLKG) and (HNLS) with small Cauchy data in some modulation spaces. Global well-posedness for fractional Schrödinger (fNLSH) equation with Hartree type nonlinearity is obtained with Cauchy data in some modulation spaces. Local well-posedness for (HNLW), (fHNLS) and (HNLKG) with rough data in modulation spaces is shown. This improves known results in Sobolev spaces in some sense. As …
In this note we discuss the conditional stability issue for the finite dimensional Calderón problem for the fractional Schrödinger equation with a finite number of measurements. More precisely, we assume that the unknown potential $q \in L^{\infty}(Ω) $ in the equation $((-Δ)^s+ q)u = 0 \mbox{ in } Ω\subset \mathbb{R}^n$ satisfies the a priori assumption that it is contained in a finite dimensional subspace of $L^{\infty}(Ω)$. Under this condition we prove Lipschitz stability estimates for the fractional Calderón problem by means of finitely many Cauchy data depending on $q$. We allow for the …
We prove the existence and a new representation formula for the gradient of the semigroup associated to an Ornstein-Uhlenbeck in a bounded convex domain in d dimensions.
Cauchy Problem for the Time-Fractional Airy-Type Equation on a Star Graph with Edge-Dependent Order of Derivative
National University of Uzbekistan（乌兹别克斯坦国立大学）；V.I. Romanovskiy Institute of Mathematics（V.I.罗曼诺夫斯基数学研究所）；University of Exact and Social Sciences in Tashkent（塔什干精确与社会科学大学）
We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such solutions near the boundary of the underlying domain.
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
In the paper, we derive an existence result for a nonlinear nonautonomous partial elliptic system on an open bounded domain with Dirichlet boundary conditions, containg fractional powers of the weak Dirichlet-Laplace operator that are meant in the Stone-von Neumann operator calculus sense. We apply a variational method which gives strong solutions of the problem under consideration.
We establish an upper bound of the sum of the eigenvalues for the Dirichlet problem of the fractional Laplacian. Our result is obtained by a subtle computation of the Rayleigh quotient for specific functions.
In the present paper, we prove time decay estimates of solutions in weighted Sobolev spaces to the second order evolution equation with fractional Laplacian and damping for data in Besov spaces. Our estimates generalize the estimates obtained in the previous studies. The second aim of this article is to apply these estimates to prove small data global well-posedness for the Cauchy problem of the equation with power nonlinearities. Especially, the estimates obtained in this paper enable us to treat more general conditions on the nonlinearities and the spatial dimension than the results in the p…
We consider an optimal control problem governed by a class of boundary value problem with the spectral Dirichlet fractional Laplacian. Some sufficient condition for the existence of optimal processes is stated. The proof of the main result relies on variational structure of the problem. To show that boundary value problem with the Dirichlet fractional Laplacian has a weak solution we employ the renowned Ky Fan Theorem.
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.
In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of mixed fractional operators with different orders include medicine, plasma physics, and population dynamics.
Local and global properties of minimal solutions for the heat equation generated by the Dirichlet fractional Laplacian negatively perturbed by Hardy's potentials on open subsets of $\R^d$ are analyzed. As a byproduct we obtain instantaneous blow-up of nonnegative solutions in the supercritical case.
In this paper we analyze the asymptotic behavior of the Dirichlet fractional Laplacian $(-Δ_{\mathbb R^{n+k}})^{s}$, with $s\in (0, 1)$, on bounded domains in $\mathbb R^{n+k}$ that become unbounded in the last $k$-directions. A dimension reduction phenomenon is observed and described via $Γ$-convergence.
In this paper, we consider the fractional Camassa-Holm equation modelling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness in Besov space $B^{s_0}_{2,1}$ with $s_0=2ν-\frac 1 2$ for $ν>\frac 3 2 $ and $s_0=\frac 5 2$ for $1<ν\leq \frac 3 2 $. Then, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time.
We consider positive solutions of a fractional Lane-Emden type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we also prove that all the known uniqueness and nondegeneracy results in the local case extend to the nonlocal regime when the fractional parameter s is sufficiently close to 1.
In this paper we firstly study the limit of minimizers of the fractional $W^{s,p}$-norms as $p\rightarrow+\infty$ in De Giorgi sense. In particular, we analyzed the $Γ$-convergence of non-homogeneous Dirichlet boundary problem for fractional $p$-Laplacian in this approximation process, and proved that as $p\rightarrow+\infty$ the minimizers of fractional $p$-Laplacian with Dirichlet boundary $Γ$-converges to a minimizer of Hölder $\infty$-Laplacian under the same Dirichlet boundary condition. On the other hand, we first investigate the asymptotic behaviour of non-homogeneous fractional $p$-fun…
Fractional-order elliptic problems are investigated in case of inhomogeneous Dirichlet boundary data. The boundary integral form is proposed as a suitable mathematical model. The corresponding theory is completed by sharpening the mapping properties of the corresponding potential operators. Also a mild condition is provided to ensure the existence of the classical solution of the boundary integral equation.
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 $\mathbb R^d$.
We are concerned in this paper with the degenerate fractional diffusion advection equations posed in bounded domains. Due to a suitable formulation, we show the existence of weak entropy solutions for measurable and bounded initial and Dirichlet boundary data. Moreover, we prove a $L^1-$type contraction property for weak entropy solutions obtained via parabolic perturbation. This is a weak selection principle which means that the weak entropy solutions are stable in this class.
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions on the Cauchy-Dirichlet boundary data, we show that the envelope of pluripotential subsolutions is semi-concave in time and continuous in space, and provides the unique pluripotential solution with such regularity.
