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Generalized solutions to functional partial differential equations of the first order. (English) Zbl 0703.35104

Zesz. Nauk. Politech. Gdańsk. 427, Mat. 14, 99 p. (1988).
The author considers existence, uniqueness and regularity of generalized solutions of certain classes of quasilinear partial differential equations of first order. In the first paragraph the author studies quasilinear hyperbolic systems in Schauder canonical form: \[ (*)\quad \sum^{n}_{j=1}A_{ij}(x,y,z(x,y))[D_ xz_ j(x,y)+ \]
\[ +\sum^{m}_{k=1}\rho_{ik}(x,y,z(x,y),(V_ 1z)(x,y),D_{y_ k}z_ j(x,y))]=f_ i(x,y,z,V_ 2z), \] i\(=1,...,n\), with boundary data on N arbitrary (not necessarily distinct) hyperplanes \(a_ k\), \(0\leq a_ k\leq a_ 0\), \(k=1,...,n\), \[ (**)\quad B_ k(y)z(a_ k,y)=\psi (y). \] Here \(A=(A_{ij})\) is a matrix-valued function defined on \(D\subset {\mathbb{R}}\times {\mathbb{R}}^ m\times {\mathbb{R}}^ n\) (where \(D=I_{a_ 0}\times {\bar \Omega}\), and \(I_{a_ 0}=[0,a_ 0]\times {\mathbb{R}}^ m\); \({\bar \Omega}=[-\Omega,\Omega]^ n\subset {\mathbb{R}}^ n\), \(\Omega \in {\mathbb{R}}_+)\), \(\rho =(\rho_{ik})\) is another matrix \(D\times \Omega^ r\to R^{nm}\), \(\psi =(\psi_ 1,...,\psi_ n): {\mathbb{R}}^ m\to {\mathbb{R}}^ n\) and \(V_ kz: I_{a_ 0}\to R^{nr}\) are Volterra type operators.
The boundary conditions (**) contains as special cases boundary conditions “à la Cesari” or of Nicoletti type. If all \(a_ k=0\) then one finds the usual Cauchy conditions.
By generalized solution the author means a function z: \(I_ a\to {\mathbb{R}}^ n\) which is absolutely continuous and satisfies almost everywhere equation (*) and (**) everywhere.
Under suitable conditions (too technical to be given here - but which amounts to good boundedness, continuity and Lipschitz conditions for A and \(A^{-1}\), for \(\rho\) and for f and which implies in particular that the matrices A and \(B_ k\) have “dominant main diagonal” in the sense of Cesari) the author shows, using the fixed point theorem in a product of two Banach spaces, the existence of generalized solutions to (*)-(**), in a closed convex subset of the space of continuous and bounded functions in \(I_ a\). Uniqueness and continuous dependence on \(\psi\) is also proved.
In the second paragraph, under slightly stronger assumptions, with a different method but which also uses a fixed point theorem, the author obtains existence and generalized solutions in a suitable space of Lipschitz functions.
In the third paragraph, the author considers mixed problems for quasilinear hyperbolic systems in two independent variables. These equations are given in diagonal form and for them local existence theorems, and also continuous dependence on boundary and initial data are given.
In the fourth paragraph the author gives sufficient conditions under which the generalized solutions considered in § 3 are classical solutions. Finally in the last paragraph, an existence theorem for the Cauchy problem for the equation \(D_ xz=F(x,y,Vz,D_ yz)\) where F is not necessarily continuous is proved by fixed point methods.
Reviewer: G.Gussi

MSC:

35L60 First-order nonlinear hyperbolic equations
35Dxx Generalized solutions to partial differential equations
35K05 Heat equation
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35L50 Initial-boundary value problems for first-order hyperbolic systems
35D05 Existence of generalized solutions of PDE (MSC2000)
47H10 Fixed-point theorems