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Local existence for solutions of fully non‐linear wave equations
Author(s) -
Lesky Peter
Publication year - 1991
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670140704
Subject(s) - mathematics , dirichlet boundary condition , boundary value problem , domain (mathematical analysis) , mathematical analysis , zero (linguistics) , scalar (mathematics) , bar (unit) , dirichlet problem , function (biology) , homogeneous , initial value problem , wave equation , mathematical physics , combinatorics , geometry , physics , philosophy , linguistics , evolutionary biology , meteorology , biology
Let Ω be a domain in ℝ n and let m ϵ ℕ; be given. We study the initial‐boundary value problem for the equation\documentclass{article}\pagestyle{empty}\begin{document}$$ F{\rm (}t,x,\bar D{\textstyle{{2m} \over x}}u{\rm (}t,x{\rm ),}\bar D{\textstyle{m \over x}}\mathop \partial \nolimits_t u{\rm (}t,x{\rm ),}\mathop \partial \nolimits_t^2 u{\rm (}t,x{\rm )) = }f{\rm (}t,x{\rm )} $$\end{document} with a homogeneous Dirichlet boundary condition; here u is a scalar function, \documentclass{article}\pagestyle{empty}\begin{document}$ \bar D_x^m u: = (\partial _x^\alpha u)_{|\alpha | \le m} $\end{document} and certain restrictions are made on F guaranteeing that energy estimates are possible. We prove the existence of a value of T >0 such that a unique classical solution u exists on [0, T ]×Ω. Furthermore, we show that T → ∞ if the data tend to zero.