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Some recent results on singular p-Laplacian equations

Guarnotta Umberto, Livrea Roberto, Marano Salvatore A.

Demonstratio Mathematica

2022

Vol. 55, nr 1

416--428

A short account of some recent existence, multiplicity, and uniqueness results for singular p-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided.

Uniqueness theorem for nonlinear hyperbolic equations with order degeneration

Semerdjieva R.

Bulletin of the Polish Academy of Sciences. Mathematics

2003

Vol. 51, no 1

99-105

Let k(y) > O, l(y) > O for y > O, k(0) = l(0) = 0; then the equation L(u) := k(y)u(xx) - (delta)y(l(y)uy) +a(x,y)ux = f (x,y,u) is strictly hyperbolic for y > O and its order degenerates on the line y = 0. We consider the boundary value problem Lu = f (x,y,u) in G, u\(AC) = 0, where G is a simply connected domain in R-2 with piecewise smooth boundary [delta]G = AB boolean Or AC boolean OR BC; AB = {(x, 0) : 0 less than or equal to x less than or equal to 1}, AC : x = F(y) = integral(0)(y)k(t)/l(t)(1/2)dt and x = 1 - F(y) are characteristic curves. It is proved that if f satisfies the Caratheodory condition and \f{x,y,z(1)}-f{x,y,z(2))\ less than or equal to C(\z(1)\(beta) + \z(2)\(beta))\z1-z2\ with some constants C > O and beta is greater than or equal to O then there exists at most one generalized solution.

On the convergence of the method of lines for quasi-nonlinear functional evolutions in Banach spaces

Jin B. J., Kim J. K.

Journal of Applied Analysis

2000

Vol. 6, nr 1

105-127

This paper is concerned with the existence of global limit solutions for the quasi-nonlinear functional evolution problem x`∈ A(t, xt) x+ G(t, xt, Ltx),t ∈ [0,T], (FDE, φ) x0=φ, where A(t, ψ 1) and G(t , ψ 1,Lt ψ 2) are defined, with respect to ψ 1, on a subspace of the space PC([-r, 0],X) of all piecewise continuous functions f : [--r, 0] → X. An appropriate subspace of PC([--r, t],X) is the domain of definition of the nonlinear operators Lt, t ∈ [0,T]. The operators A(t, ψ)x are w-dissipative and Lipschitz - like in (t, ψ ) which are more general conditions than those of Karsatos-Liu. The operators G and Lt are Lipschitzian mappings on their respective domains. Moreover, we investigate the uniqueness and strong solution for such problem.

Existence of strong solutions to stochastic differential equations in conuclear spaces

Jakubowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 3

247-252

The existence of an unique strong solutions to stochastic differential equations with respect to a generalized non-homogeneous Wiener process in the dual of a nuclear space is proved under monotonicity condition and conditions which guarantee e~stence of weak solutions.