Global solvability of the mixed problem for first order functional partial differential equations. (English) Zbl 0729.35139

Turo, Jan

Ann. Pol. Math. 52, No. 3, 231-238 (1991).

Global generalized solutions of the systems \[ (1)\quad D_ xz_ i(x,y)+\lambda_ i(x,y,z(x,y),(Vz)(x,y))D_ yz_ i(x,y)=f_ i(x,y,z(x,y),(Vz)(x,y)), \] (x,y)\(\in \{(x,y):\) \(0\leq x\leq a_ 0\), \(0\leq y\leq b\}\), \(i=1,...,n\) with the initial conditions \[ (2)\quad z(0,y)=\phi (y),\quad y\in [0,b], \] and the boundary conditions \[ (3)\quad z_ i(x,0)=h_{0i}(x),\quad i\in \{i:\;sgn \lambda_ i(0,0,0,0)=1\}, \] \[ (4)\quad z_ i(x,b)=h_{bi}(x),\quad i\in \{i:\;sgn \lambda_ i(0,b,0,0)=-1\},\quad x\in [0,a_ 0], \] are investigated. The global solvability of problem (1)-(4) is ensured by the monotonic behavior of given functions and a growth restriction on the right-hand side. The functional operator V includes retarded arguments and Volterra hereditary operators.

Reviewer: Jan Turo

MSC:

35R10	Functional partial differential equations
35D05	Existence of generalized solutions of PDE (MSC2000)
35L60	First-order nonlinear hyperbolic equations

Keywords:

mixed problem; hyperbolic system; Global generalized solutions; global solvability; Volterra hereditary operators