517.956Using the method of contracting mappings, we prove, for small values of time, the existence and uniqueness of a generalized Lipschitz solution of a mixed problem with unknown boundaries for a hyperbolic quasilinear system of first-order equations represented in terms of Riemann invariants with nonlocal (nonseparated and integral) boundary conditions.
Abstract:We consider the inverse problem of determining how the physiological structure of a harvested population evolves in time, and of finding the time-dependent effort to be expended in harvesting, so that the weighted integral of the density, which may be, for example, the total number of individuals or the total biomass, has prescribed dynamics. We give conditions for the existence of a unique, global, weak solution to the problem. Our investigation is carried out using the method of characteristics and a generalization of the Banach fixed-point theorem.
517.956Using the method of characteristics and the method of contracting mappings, we establish the local classical solvability of a problem for a hyperbolic system of quasilinear first-order equations with moving boundaries and nonlinear boundary conditions. Under additional assumptions on the monotonicity and sign constancy of initial data and a restriction on the growth of the right-hand sides of the system, we formulate sufficient conditions for the global classical solvability of the problem.
Statement of the Problemk , k = 1, 2, we consider the system of quasilinear partial differential equationswhere the functions λ i , f i : R × [0, T ] × R n → R are given. LetThe sets I 1 and I 2 may intersect, and I 1 ∪ I 2 not necessarily coincides with {1, . . . , n}. We introduce the initial and boundary conditionswhere
Local Solvability of the ProblemWe introduce the notation
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