Classical solutions of hyperbolic partial differential equations with implicit mixed derivative

• Autorzy: Salvatore A. Marano
• Języki publikacji: EN

Abstrakty

EN

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).

Słowa kluczowe

EN

hyperbolic equation; implicit mixed derivative; classical solution

Kategorie tematyczne

• 35L15: Initial value problems for second-order hyperbolic equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1991-1992
• Tom: 56
• Numer: 2
• Strony: 163-178

Daty

wydano

1992

otrzymano

1990-10-25

poprawiono

1991-04-15

Twórcy

autor

Salvatore A. Marano

• Dipartimento di Matematica, Città Universitaria, Viale A. Doria, 6 I-95125 Catania, Italy

Bibliografia

• [1] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math. 60, Marcel Dekker, 1980.
• [2] J. Bryszewski, L. Górniewicz and T. Pruszko, An application of the topological degree theory to the study of the Darboux problem for hyperbolic equations, J. Math. Anal. Appl. 76 (1980), 107-115.
• [3] G. Emmanuele and B. Ricceri, Sull'esistenza delle soluzioni delle equazioni differenziali ordinarie in forma implicita negli spazi di Banach, Ann. Mat. Pura Appl. (4) 129 (1981), 367-382.
• [4] P. Hartman and A. Wintner, On hyperbolic partial differential equations, Amer. J. Math. 74 (1952), 834-864.
• [5] B. Rzepecki, On the existence of solutions of the Darboux problem for the hyperbolic partial differential equations in Banach spaces, Rend. Sem. Mat. Univ. Padova 76 (1986), 201-206.
• [6] G. Vidossich, Hyperbolic equations as ordinary differential equations in Banach space, preprint S.I.S.S.A.