Narrow Results

In this work, we explore a new class of diffusion systems involving multi-term fractional integral operators and nonlinear coupling terms in a Hilbert space. First, we use a time semi-discrete approach grounded in the backward Euler difference formulation (i.e. Rothe’s method) to introduce a discrete iterative system. Then, the existence and uniqueness as well as the priori estimations of solution for the discrete iterative system are proved. Moreover, an approximating parabolic coupled system is considered, and an convergence result which shows that the solution of approximating coupled system converges to the unique strong solution of original problem, is obtained. Finally, we give three examples to illustrate the validity of the theoretical results.

We establish global weighted W^2,p, 2 < p < ∞, estimates for the solution to the Dirichlet problem for an elliptic equation in nondivergence form with BMO coefficients in a C^1,1 domain under the assumption that the matrix of the coefficients has a small BMO semi-norm while the associated weight belongs to a Muckenhoupt class. These conditions are weaker than those reported in the literature.

This paper is concerned with a blow-up criterion of strong solutions for three-dimensional compressible isentropic magnetohydrodynamic equations with vacuum. It is shown that if the density and velocity satisfy , where , 3 < r ≤ ∞ and denotes the weak L^r-space, then the strong solutions to the Cauchy problem of the compressible magnetohydrodynamic equations can exist globally over [0, T].

This paper analyzes the superlinear indefinite prescribed mean curvature problem

We prove existence and pathwise uniqueness results for four different types of stochastic differential equations (SDEs) perturbed by the past maximum process and/or the local time at zero. Along the first three studies, the coefficients are no longer Lipschitz. The first type is the equation

In this paper we prove well-posedness and stability of a class of stochastic delay differential equations with singular drift. Moreover, we show local well-posedness under localized assumptions.

In this paper, we study the conditions for the existence of strong solutions (both local and global) for stochastic bidomain equations. To this end, we use a priori energy estimates and Serrin-type theorems. We further address the asymptotic behavior of the solutions, which includes the analysis of small stochastic perturbations and large deviations. In a separate section we specify the support of the invariant measure, whose existence was established in [M. Hieber, O. Misiats and O. Stanzhytskyi, On the bidomain equations driven by stochastic forces, Discrete Contin. Dyn. Syst. 40 (11) (2020) 6159–6177].

We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean–Vlasov equations with random non-Lipschitz continuous coefficients. In the deterministic case, we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada–Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise exponential stability. Furthermore, Lyapunov exponents are computed explicitly.

We study the density-dependent incompressible Cahn–Hilliard–Navier–Stokes system, which describes a two-phase flow of two incompressible fluids with different densities. We establish the local existence and uniqueness of strong solutions to the initial value problem in a bounded domain, when the initial density function enjoys a positive lower bound.

This paper presents new results for strong solutions and their coincidence sets of the obstacle problem for linear hyperbolic operators of first order. An inequality similar to the Lewy–Stampacchia ones for elliptic and parabolic problems is shown. Under nondegeneracy conditions the stability of the coincidence set is shown with respect to the variation of the data and with respect to approximation by semilinear hyperbolic problems. These results are applied to the asymptotic stability of the evolution problem with respect to the stationary coercive problem with obstacle.

In this paper, by using classical Faedo–Galerkin approximation and compactness method, the existence of martingale solutions for the stochastic 3D Navier–Stokes equations with nonlinear damping is obtained. The existence and uniqueness of strong solution are proved for β > 3 with any α > 0 and $\alpha \ge \frac{1}{2}$ as β = 3. Meanwhile a small time large deviation principle (LDP) for the stochastic 3D Navier–Stokes equations with damping is obtained for β > 3 with any α > 0 and $\alpha \ge \frac{1}{2}$ as β = 3.

This paper presents new results for strong solutions and their coincidence sets of the obstacle problem for linear hyperbolic operators of first order. An inequality similar to the Lewy–Stampacchia ones for elliptic and parabolic problems is shown. Under nondegeneracy conditions the stability of the coincidence set is shown with respect to the variation of the data and with respect to approximation by semilinear hyperbolic problems. These results are applied to the asymptotic stability of the evolution problem with respect to the stationary coercive problem with obstacle.

In a real Hilbert space we study the existence, uniqueness and asymptotic behaviour of the strong and weak solutions to a class of nonlinear differential systems subject to some extreme conditions and initial data. For the proofs of our theorems we use several results from the theory of monotone operators and of nonlinear evolution equations of monotone type.

The ABP type maximum principle for L^p-viscosity solutions of fully nonlinear second order elliptic/parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is exhibited.