Classical solutions of hyperbolic partial differential equations with implicit mixed derivative
Annales Polonici Mathematici, Tome 56 (1991) no. 2, pp. 163-178
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
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).
Keywords:
hyperbolic equation, implicit mixed derivative, classical solution
Affiliations des auteurs :
Salvatore Marano 1
@article{10_4064_ap_56_2_163_178,
author = {Salvatore Marano},
title = {Classical solutions of hyperbolic partial differential equations with implicit mixed derivative},
journal = {Annales Polonici Mathematici},
pages = {163--178},
year = {1991},
volume = {56},
number = {2},
doi = {10.4064/ap-56-2-163-178},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/ap-56-2-163-178/}
}
TY - JOUR AU - Salvatore Marano TI - Classical solutions of hyperbolic partial differential equations with implicit mixed derivative JO - Annales Polonici Mathematici PY - 1991 SP - 163 EP - 178 VL - 56 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4064/ap-56-2-163-178/ DO - 10.4064/ap-56-2-163-178 LA - en ID - 10_4064_ap_56_2_163_178 ER -
%0 Journal Article %A Salvatore Marano %T Classical solutions of hyperbolic partial differential equations with implicit mixed derivative %J Annales Polonici Mathematici %D 1991 %P 163-178 %V 56 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4064/ap-56-2-163-178/ %R 10.4064/ap-56-2-163-178 %G en %F 10_4064_ap_56_2_163_178
Salvatore Marano. Classical solutions of hyperbolic partial differential equations with implicit mixed derivative. Annales Polonici Mathematici, Tome 56 (1991) no. 2, pp. 163-178. doi: 10.4064/ap-56-2-163-178
Cité par Sources :