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We consider a perforated domain Ω(ε) of R² with a small hole of size ε and we study the behavior of the solution of a mixed Neumann-Robin problem in Ω(ε) as the size ε of the small hole tends to 0. In addition to the geometric degeneracy of the problem, the nonlinear ε-dependent Robin condition may degenerate into a Neumann condition for ε = 0 and the Robin datum may diverge to infinity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as ε tends to 0 and to understand how the boundary condition affects the behavior of the solutions when ε is close to 0. The present paper extends to the planar case the results of [36] dealing with the case of dimension n ≥ 3. [ABSTRACT FROM AUTHOR]

We study the equivalences and the implications between linear (or homogeneous) nth order fractional differential equations (FDEs) and integral equations in the spaces L¹ (a, b) and C[a, b] when n ≥ 2. We establish the equivalence in C[a, b] of the IVP of the nth order FDE subject to the initial condition u(i) (a) = ui for i ∈ {0, 1, . . ., n − 1} when n ≥ 2. The difficulty is that the known conditions for such equivalence for the first order FDEs are not sufficient for equivalence in the nth order FDEs with n ≥ 2. In this article we provide additional conditions to ensure the equivalence for the nth order FDEs with n ≥ 2. In particular, we obtain conditions under which solutions of the integral equations are solutions of the linear nth order FDEs. These results are keys for further studying the existence of solutions and nonnegative solutions to initial and boundary value problems of nonlinear nth order FDEs. This is done via the corresponding integral equations by topological methods such as the Banach contraction principle, fixed point index theory, and degree theory. [ABSTRACT FROM AUTHOR]

We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on R^d. Using the properties of the corresponding semiflow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and exponential integrability of the traveling wave profiles.

In this article, we considered two inverse source problems for fourth-order parabolic differential equation with fractional derivative in time. Determination of a space dependent source term from the data given at some time t=T is considered in one problem while other addresses the recovery of a time dependent source term from the integral type over-determination condition. Existence and uniqueness of the solution of both inverse source problems are proved. The stability results for the inverse problems are presented.

We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on ℝd. Using the properties of the corresponding semi ow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and expo- nential integrability of the traveling wave profiles. [ABSTRACT FROM AUTHOR]

We prove the unique solvability of a boundary-value problems for a third-order loaded integro-differential equation with variable coefficients, by reducing the equation to a Volterra integral equation.

We establish the unique solvability for an inverse problem for semi-linear fractional telegraph equation $$ D^\alpha_t u+r(t)D^\beta_t u-\Delta u=F_0(x,t,u,D^\beta_t u), \quad (x,t) \in \Omega_0\times (0,T] $$ with regularized fractional derivatives $D^\alpha_t u, D^\beta_t u$ of orders $\alpha\in (1,2)$, $\beta\in (0,1)$ with respect to time on bounded cylindrical domain. This problem consists in the determination of a pair of functions: a classical solution $u$ of the first boundary-value problem for such equation, and an unknown continuous coefficient $r(t)$ under the over-determination condition $$ \int_{\Omega_0}u(x,t)\varphi(x)dx=F(t), \quad t\in [0,T] $$ with given functions $\varphi$ and $F$.

In this article, we considered two inverse source problems for fourth-order parabolic differential equation with fractional derivative in time. Determination of a space dependent source term from the data given at some time t = T is considered in one problem while other addresses the recovery of a time dependent source term from the integral type over-determination condition. Existence and uniqueness of the solution of both inverse source problems are proved. The stability results for the inverse problems are presented. [ABSTRACT FROM AUTHOR]

We establish the unique solvability for an inverse problem for semi- linear fractional telegraph equation Dtα u + r(t) Dtβ u - Δu = Fo(x,t,u, Dtβ u), (x,t) ∈ Ω0 x (0,T] with regularized fractional derivatives Dtαu, Dtβu of orders α ∈ (1,2), β ∈ (0,1) with respect to time on bounded cylindrical domain. This problem consists in the determination of a pair of functions: a classical solution u of the first boundary-value problem for such equation, and an unknown continuous coefficient r(t) under the over-determination condition ... with given functions φ and F. [ABSTRACT FROM AUTHOR]