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Classical solutions of hyperbolic partial differential equations with implicit mixed derivative

100%

Marano S.

Annales Polonici Mathematici

1991-1992

tom 56

nr 2

163-178

Let f be a continuous function from $[0,a] × [0,β] × (ℝ^n)⁴$ into $ℝ^n$. Given $u₀,v₀ ∈ C⁰([0,β],ℝ^n)$, with f(0, x, ∫_0^x u₀(s)ds, ∫_0^x v₀(s)ds, u₀(x), v₀(x)) = v₀(x) for every x ∈ [0,β], consider the problem (P) { ∂²z/(∂t∂x) = f(t, x, z, ∂z/∂t, ∂z/∂x, ∂²z/(∂t∂x)), $z(t,0) = ϑ_{ℝ^n}$, $z(0,x)=∫_0^x u₀(s)ds$, ∂²z(0,x)/(∂t∂x) = v₀(x). In this paper we prove that, under suitable assumptions, problem (P) has at least one classical solution that is local in the first variable and global in the other. As a consequence, we obtain a generalization of a result by P. Hartman and A. Wintner ([4], Theorem 1).

Semilinear evolution equations of the parabolic type

100%

Bochenek J.

Annales Polonici Mathematici

1999

tom 72

nr 3

197-206

This paper is devoted to the investigation of the abstract semilinear initial value problem du/dt + A(t)u = f(t,u), u(0) = u₀, in the "parabolic" case.

Ograniczanie wyników

2 Annales Polonici Mathematici

1 Bochenek J.

1 Marano S.

1 1999

1 1992

Wyniki wyszukiwania

Classical solutions of hyperbolic partial differential equations with implicit mixed derivative

Semilinear evolution equations of the parabolic type