We consider a parabolic-type PDE with diffusion given by fractional Laplacian operator and quadratic nonlinearity of the 'gradient' solution, convoluted singular term b. Our first result is well-posedness for this problem: show existence uniqueness (local in time) mild solution. The main about blow-up said particular we find sufficient conditions on initial datum b to ensure solution finite time.

This paper focuses on the study of existence a mild solution to time and space-fractional stochastic equation perturbed by multiplicative white noise. The required results are obtained analysis techniques, fractional calculus, semigroup theory Leray-Schauder nonlinear alternative.

in this paper, we prove the existence and uniqueness results to the random fractional functional differential equations under assumptions more general than the lipschitz type condition. moreover, the distance between exact solution and appropriate solution, and the existence extremal solution of the problem is also considered.

We study almost automorphic (mild) solutions of the semilinear fractional equation ∂ t u = Au + ∂ α−1 t f(·, u), 1 < α < 2, considered in a Banach space X, where A is a linear operator of sectorial type ω < 0. We prove the existence and uniqueness of an almost automorphic mild solution assuming f(t, x) is almost automorphic in t for each x ∈ X, satisfies some Lipschitz type conditions and takes...

Abstract. In this paper we discuss the existence of PC-mild solutions for Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving Caputo fractional derivative. By utilizing the theory of operators semigroup, probability density functions via impulsive conditions, a new concept on a PC-mild solution for our problem is introduced. Our main techniques based on ...

In this paper, using the fixed point theory in cone metric spaces, we prove the existence of a unique solution to a first-order ordinary differential equation with periodic boundary conditions in Banach spaces admitting the existence of a lower solution.

Abstract In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator F (t, Y, Z) has a quadratic growth in Z. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then appl...

This is a study of the existence of bifurcation branches for the problem of finding even, periodic solutions in fourth-order, reversible Hamiltonian systems such that the Hamiltonian evaluates to zero along each solution on the branch. The class considered here is a generalisation of both the Swift-Hohenberg and extended Fisher-Kolmogorov equations that have been studied in several recent paper...