The paper deals with homogenisation problems for high-contrast symmetric convolution-type operators with integrable kernels in media with a periodic microstructure. We adapt the two-scale convergence method to nonlocal convolution-type operators and obtain the homogenisation result both for problems stated in the whole space and in bounded domains with the homogeneous Dirichlet boundary condition. Our main focus is on spectral analysis. We describe the spectrum of the limit two-scale operator and characterise the limit behaviour of the spectrum of the original problem as the microstructure period tends to zero. It is shown that the spectrum of the limit operator is a subset of the limit of the spectrum of the original operator, and that they need not coincide.

Abstract We investigate density results for solutions of the non-local heat equation at a fixed time slice, considering two models: one with homogeneous Dirichlet boundary conditions and another with singular boundary data. In both cases, the non-local exponent satisfies $a \in (\tfrac{1}{2}, 1)$. We study both qualitative and quantitative approximation properties.
 
For the model with singular boundary data, we assume the potential is non-negative, sufficiently small, and exhibits mild growth. The smallness condition is explicit and depends only on the domain and the spatial dimension.
 
We also address Calderón-type inverse problems for these parabolic models, recovering the potential from solution data measured either on the boundary or at a fixed time slice. The Pohozaev identity is a key tool in establishing both the density results and the inverse problem analysis.
 
Finally, we apply the Pohozaev identity to an elliptic eigenvalue problem and show that the corresponding eigenfunctions, when divided by a suitable power of the distance to the boundary, cannot vanish on any non-empty open subset of the boundary. This result holds without restrictions on the non-local exponent.

In this paper, we investigate the isoparametric finite element semidiscretization for a parabolic problem on a curvilinear polyhedral domain Ω ⊆ R N \Omega \subseteq \mathbb {R}^N with homogeneous Dirichlet boundary condition. The domain Ω \Omega may include nonconvex corners, i.e., with edge openings possibly greater than π \pi . We establish the analyticity and maximal regularity of the discrete semigroup by employing a transformation method to address the domain perturbation effect Ω ≠ Ω h \Omega \neq \Omega _h . As an application of the logarithmically quasi-maximal L ∞ L^\infty -regularity, we derive a quasi-optimal maximum-norm error estimate for the semidiscrete isoparametric finite element methods on curvilinear polyhedral domains with edge openings smaller than π \pi , which includes a term of quasi-optimal order due to domain perturbation.

The purpose of this paper is to study an isotropic problem with non homogeneous Dirichlet boundary condition and L_1 data in a variable-exponent Sobolev space. We start by showing the existence and uniqueness of the weak solution when the source term is bounded. Finally, we use a problem-based approach to prove the existence and uniqueness of entropy solution when the source term is integrable.

We investigate the initial-boundary value problem for the nonlinear viscoelastic Kirchhoff-type wave equation\begin{equation*}u_{tt} - \left(1 + \|\nabla u(t)\|^2 \right) \Delta u + \int_0^t g(t - s) \Delta u(s) \, ds + \alpha u_t = |u|^{p-1} \ln |u|, \quad x \in \Omega, t > 0,\end{equation*}with homogeneous Dirichlet boundary conditions. Local existence and uniqueness of weak solutions are established via a Banach fixed-point argument. Moreover, we prove a finite-time blow-up result: if the initial energy is positive and the data satisfy a compatibility condition, the solution’s $H_0^1$-norm becomes unbounded in finite time. This work extends existing results by capturing the combined effects of Kirchhoff-type nonlinearity, viscoelastic damping, and a logarithmic source.

Abstract We prove global second-order regularity for a class of quasilinear elliptic equations, both with homogeneous Dirichlet and Neumann boundary conditions. A condition on the integrability of the second fundamental form on the boundary of the domain is required. As a consequence, with the additional assumption that the source term has a sign, we obtain integrability properties of the inverse of the gradient of the solution. Assuming convexity of the domain, no boundary regularity is required.

We study the second order elliptic equations of non-divergence form in a planar domain with complicated geometry. In this case the domain winds around a fixed circle infinitely many times and converges to it when the rotating angle goes to infinity. For the homogeneous equation and the homogeneous Dirichlet boundary condition, in the case of bounded drifts, we prove that the maximum of the solution on the cross-section corresponding to a given rotating angle either grows or decays exponentially as the angle goes to infinity. Results for the oscillation and its asymptotic estimates are also obtained for inhomogeneous Dirichlet data. If the drift is unbounded but does not grow to infinity too fast, then the above maximum also goes to either zero or infinity. For the inhomogeneous equation, we obtain the estimates in the case of bounded forcing functions. Moreover, we establish the uniqueness of the solution and its continuous dependence on the boundary data and the forcing function.

This paper investigates the existence of nonnegative weak solutions to a class of degenerate elliptic equations with singular nonlinearities. The problem under consideration is of the form [Formula: see text] with homogeneous Dirichlet boundary conditions, where [Formula: see text] is a bounded domain, [Formula: see text], [Formula: see text], [Formula: see text] is a nonnegative element of the dual Sobolev space [Formula: see text], and [Formula: see text] is a continuous function that may blow up at zero but remains bounded at infinity. The degeneracy of the principal part, controlled by the parameter [Formula: see text], adds significant difficulty to the analysis. Using a double approximation scheme (regularizing both the degeneracy and the singularity), truncation arguments, monotonicity methods, and the Schauder fixed point theorem, we establish the existence of a solution [Formula: see text] under appropriate conditions on the data. Our main contribution lies in the simultaneous treatment of degeneracy and singularity, extending classical results to a broader class of non-uniformly elliptic operators. The proofs rely on uniform a priori estimates, compactness arguments, and a careful passage to the limit in the approximate problems.

We study an inverse source problem for a semilinear diffusion equation involving a Caputo-type time-fractional derivative whose order is a function of time. The equation is considered in a bounded Lipschitz domain Ω⊂Rd, d≥1, and is supplemented with homogeneous Dirichlet boundary conditions. The source term is taken to be separable, h(t)f(x), where the temporal component h(t) is unknown. This quantity is to be identified from spatially localized measurements m(t) of the solution. In this setting, we establish existence and uniqueness results in suitable function spaces, thereby demonstrating the well-posedness of the corresponding inverse source problem.

This work investigates the global well-posedness and long-term dynamics of two-component reaction-diffusion systems on bounded domains under homogeneous Dirichlet boundary conditions. We introduce a weaker dissipative condition that enables us to prove the global existence and uniqueness of classical solutions to the associated Cauchy problem, without imposing any growth constraints on the nonlinear terms. The admissible nonlinearities include, but are not limited to, polynomial and exponential growth types. Furthermore, we demonstrate that such systems admit both global and exponential attractors, which exhibit finite-dimensional characteristics in appropriate continuous function spaces.

In this work, we investigate the global asymptotic behavior of an age-structured epidemic model with diffusion, under Dirichlet boundary conditions and a protection phase of limited duration. We consider a bounded n-dimensional spatial domain with homogeneous Dirichlet boundary conditions. The system is directly transformed to a coupled system of reaction-diffusion equations and a continuous difference equation with a time-delay and a non-local spatial term due to mobility during the protection phase. The Basic Reproduction Number (BRN) is determined for this system and some properties are obtained according to this parameter. Using Schauder's fixed point theorem, it is shown that disease-free equilibrium always exists and is globally attractive when BRN is less than one. Otherwise, we use the method of monotonic iterations for a particular elliptic problem to show that the endemic steady-state exists when BRN is greater than one. This condition on BRN follows naturally from the construction of an upper- and a lower-solution. Moreover, we prove that the system is uniformly strongly persistent. Finally, we perform numerical simulations that confirm and complete our theoretical results.

This paper presents the inverse problem (IP) for the fractional order two-dimensional parabolic diffusion equation (FOTDPDE) formulated to depend on a initial-boundary value problem (IBVP) with homogeneous Dirichlet boundary conditions (DBC). The model involves a fractional-order Caputo derivative (FOCD) and an inverse time-dependent source term. A Crank-Nicholson finite difference scheme (CN-FDS) is constructed, and stability inequalities and a theorem in the discrete $ L^2 $ norm are proved to ensure unconditional stability of the proposed scheme. Results calculated by using finite difference methods (FDM) have a temporal convergence rate of $ O(\tau^{2-\alpha}) $ and second-order spatial accuracy. Numerical examples are tested to confirm the theoretical stability results and to represent the effectiveness and the accuracy of the method for solving IP for FOTDPDE depending on BVP.

We study an elliptic equation, with homogeneous Dirichlet bound-ary conditions, driven by a mixed type operator (the sum of the Lapla-cian and the fractional Laplacian), involving a parametric reaction and an undetermined source term. Applying a recent abstract critical point theorem of Ricceri, we prove existence of a solution for a convenient source and small enough parameters.

ABSTRACT In this paper, we investigate the global dynamics of a two‐species competition reaction–diffusion model in a time‐varying domain under the homogeneous Dirichlet and Neumann boundary conditions. Under appropriate conditions, we establish the competitive exclusion principle for asymptotically bounded and periodic domains, respectively. By the method of upper and lower solutions and comparison arguments, we prove that one species will exclude the other in an asymptotically unbounded domain. We further apply the analytic results to a Lotka–Volterra competition model for its global dynamics and conduct numerical simulations to illustrate our findings.

In this paper, we prove the lower bounds of the blow-up time of solutions for certain Caputo time fractional diffusion equations and systems with nonlinear memory terms under homogeneous Dirichlet boundary conditions. The proofs of our results rely on the auxiliary function method, the differential inequality technique, and the properties of the solutions of fractional differential inequalities.

This study investigates a prey-taxis system with a functional response reflecting the cooperation effects proposed by [Cosner et al.; 1999]. Under homogeneous Dirichlet boundary conditions, the existence and nonexistence of coexistence states and global bifurcation branches are examined. Our mathematical analysis relies on a priori bounds, principal eigenvalue theory, homogenization technique and global bifurcation method. This leads us to study the coexistence region and compare our results to the prey-taxis system with linear functional response. Our results suggest that the new functional response reduces the likelihood of predator–prey coexistence.

Abstract This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which transfers the original model to an equivalent but more tractable form, the uniqueness of solutions to the variable-exponent subdiffusion equation with homogeneous Dirichlet boundary condition are established from its partial information, which results in the uniqueness of the inverse space-dependent source problem from local internal observation or partial boundary flux data. Then, based on the variational identity connecting the inversion input data with the unknown source function, we propose a weak norm and prove the conditional stability for the inverse problem in this norm. The iterative thresholding algorithm and Nesterov iteration scheme are employed to numerically reconstruct the smooth and non-smooth sources, respectively. Numerical experiments are performed to investigate their effectiveness.

ABSTRACT The fractional generalized Davey–Stewartson System (FGDSS) is an extension of the classical Davey–Stewartson model, introducing Riesz fractional derivatives to capture nonlocal interactions and anomalous dispersion effects. Incorporating fractional operators enhances the physical realism of multidimensional wave descriptions, but it also poses substantial difficulties for numerical discretization and stability control. In this study, we develop a fully discrete finite‐difference scheme that employs the Crank–Nicolson (CN) approach for temporal integration together with Ortigueira's centered‐difference method for approximating Riesz fractional derivatives in space. The proposed algorithm attains second‐order accuracy in both time and space while maintaining a discrete counterpart of the Hamiltonian structure. To resolve the nonlinear coupling, we adopt an iterative Picard strategy at each time increment. A key contribution of this work is the demonstration that, under homogeneous Dirichlet boundary conditions, the method preserves discrete analogs of key invariants such as mass, energy, and momentum. Through extensive numerical tests—including scenarios featuring soliton dynamics—we validate the accuracy, robustness, and long‐time stability of the scheme. These results highlight the potential of the CN‐Ortigueira formulation as a practical tool for analyzing complex fractional dispersive models and pave the way for future developments involving adaptive meshes and higher‐order discretizations.

This work investigates the global well-posedness and long-term dynamics of two-component reaction-diffusion systems on bounded domains under homogeneous Dirichlet boundary conditions. We introduce a weaker dissipative condition that enables us to prove the global existence and uniqueness of classical solutions to the associated Cauchy problem, without imposing any growth constraints on the nonlinear terms. The admissible nonlinearities include, but are not limited to, polynomial and exponential growth types. Furthermore, we demonstrate that such systems admit both global and exponential attractors, which exhibit finite-dimensional characteristics in appropriate continuous function spaces.

Let [Formula: see text], [Formula: see text], be a bounded connected open set with Lipschitz boundary. We consider the weighted eigenvalue problem [Formula: see text] in [Formula: see text] with [Formula: see text], [Formula: see text] and with homogeneous Dirichlet and Robin boundary conditions. First, we study weak* continuity, convexity and Gâteaux differentiability of the map [Formula: see text], where [Formula: see text] is the principal eigenvalue. Then, denoting by [Formula: see text] the class of rearrangements of a fixed weight [Formula: see text] and assuming that [Formula: see text] is positive on a set of positive Lebesgue measure, we investigate the minimization and maximization of [Formula: see text] over [Formula: see text]. The minimization problem has been already discussed in some papers; here we give an alternative treatment of some known results about the existence and characterization of minimizers of [Formula: see text]. We underline that our approach allows us to deal with Dirichlet and Robin boundary conditions together. Instead, to our best knowledge, the maximization problem has been only partially addressed in the literature; it turns out that the maximization of [Formula: see text] is more intricate than its minimization. In our work we discuss existence, uniqueness and characterization of maximizers both in [Formula: see text] and in its weak* closure [Formula: see text]. In particular, we provide an original full description of the unique maximizer in the case of Dirichlet boundary conditions. To prove this result we give a generalization of a lemma of Burton [Rearrangements of functions, maximization of convex functionals and vortex rings, Math. Ann. 276 (1987) 225–253, doi:10.1007/bf01450739] about rearrangement of functions, which we did not find in literature. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favorable and unfavorable habitats in order to increase the chances of survival or extinction of a population.