For the Neumann problem, we prove that the solutions reach the average of its initial datum in finite time.
Under additional assumptions, we prove Gâteaux and Fréchet differentiability of solutions with respect to the initial datum.
We consider the nonlinear nonelliptic Schrödinger equation defined by i∂tu+(∂x2−∂y2)u+γ|u|2u="0 with initial datum in L2(R2).
We prove existence, uniqueness and Lipschitz dependence on the initial datum for mild solutions of stochastic partial differential equations with Lipschitz coefficients driven by Wiener and Poisson noise.
We prove existence and uniqueness of this type of solutions when the initial datum is locally integrable (for 1
We prove that the mild solution to a semilinear stochastic evolution equation on a Hilbert space, driven by either a square integrable martingale or a Poisson random measure, is (jointly) continuous, in a suitable topology, with respect to the initial datum and all coefficients.
Denoting by λ∈(−∞,2]⧹{0} the degree of homogeneity of the coagulation kernel a, measure-valued solutions are shown to be unique under the sole assumption that the moment of order λ of the initial datum is finite. A similar result was already available for the kernels a x,y)= 2, x+y and xy, and is extended here to a much wider class of kernels by a different approach.
For arbitrary initial datum.
The resulting system effectively solves the zero-dispersion KdV with an arbitrary initial datum.
Assume, if is a subsolution of (1.1)–(1.3) corresponding to the initial datum, and is a supersolution of (1.1)–(1.3) corresponding to the initial datum, then.
Now let us study the regular dependence of the solution on the initial datum.
