We introduce a new class of Backward Stochastic Differential Equations in which the T -terminal value YT of the solution (Y, Z) is not fixed as a random variable, but only satisfies a weak constraint of the form E[Ψ(YT )] ≥ m, for some (possibly random) non-decreasing map Ψ and some threshold m. We name them BSDEs with weak terminal condition and obtain a representation of the minimal time t-va...

The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration schem...

Let Ω be a domain in Rd, d ≥ 2, and 1 < p <∞. Fix V ∈ Lloc(Ω). Consider the functional Q and its Gâteaux derivative Q′ given by Q(u) := ∫ Ω (|∇u|+V |u|)dx, 1 p Q(u) := −∇·(|∇u|∇u)+V |u|u. If Q ≥ 0 on C∞ 0 (Ω), then either there is a positive continuous function W such that ∫ W |u|p dx ≤ Q(u) for all u ∈ C∞ 0 (Ω), or there is a sequence uk ∈ C ∞ 0 (Ω) and a function v > 0 satisfying Q ′(v) = 0, ...

We are dealing with a reaction-diffusion equation ut = ∆u+uyy+f(u) in R , where (x, y) = (x1, . . . , xn, y) ∈ R n+1 and ∆ is the Laplacian in R. Suppose that the equation has a bistable nonlinearity, namely it has two stable constant solutions u = 0, 1 and an unstable one between those. With the unbalanced condition ∫ 1 0 f(u)du > 0 the equation allows planar traveling waves connecting two con...

Abstract Weak solutions of a Vlasov-Schrödinger-Poisson system are shown to exist in the stationary and time-dependent situations. This system models the transport and interaction of electrons in a bidimensional electron gas. The particles are assumed to have a wave behaviour in the confinement directions (z) and to behave like point particles in the directions parallel to the electron gas (x)....

We study a spatially-explicit harvesting model in periodic or bounded environments. The model is governed by a parabolic equation with a space-dependent nonlinearity of KPP type, and a negative external forcing term. The domain is either the whole space R , with periodic coefficients, or a bounded domain. Analyzing the stationary states, we define two main types of solutions: the “significant” ...

Many papers and books on fractional differential equations have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function see, e.g., 1, 2 and to problems of analyticity in the complex domain 3 . Moreover, Delbosco and Rodino 4 considered the existence of a solution for the nonlinear fractional differential equation D 0 u f t,...

We formulate and study a one-dimensional single-species di¬usive-delay population model. The time delay is the time taken from birth to maturity. Without di¬usion, the delay di¬erential model extends the well-known logistic di¬erential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple globa...

In this paper, using sub-supersolution argument, we prove an existence result on positive solution for an ecological model under certain conditions. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. The assumptions are that the ecosystem is spatially homogeneous and the herbivore density is a constant which are valid assumptions for mana...

The Cauchy problem for a nonlinear diffusion-convection equation is studied. The equation may be classified as being of degenerate parabolic type with one spatial derivative and a time derivative. It is shown that under certain conditions solutions of the initial-value problem exhibit instantaneous shrinking. This is to say, at any positive time the spatial support of the solution is bounded ab...