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Strong and weak non-linear parabolic differential-functional inequalities with functional boundary conditions. (English) Zbl 0563.35083

Let G be an open and bounded set in \(R^ n\). Let be \(D=[0,T)\times G\), \(\Sigma_ 0=\{0\}\times \bar G\), \(\Gamma =(0,T)\times \partial G\). Assume that \(\Gamma =\Sigma \cup \Sigma^*\), where \(\Sigma\) denotes the set of all points (t,x)\(\in \Gamma\) which can be attained from D by some segment \(\ell (t,x)\) orthogonal to the t-axis. Assuming that w is in a certain class of admissible functions, the author considers an interior operator \(Pw(t,x)w_ t(t,x)-f(t,x,w(t,x),w_ x(t,x),w_{xx}(t,x),w),\) (t,x)\(\in Int D\), and a boundary operator \(Rw(t,x)=w(t,x)-\phi (t,x,w(t,x),\partial w/\partial \ell (t,x),w),\) (t,x)\(\in \Sigma\). These operators are similar to those considered by R. Redheffer and W. Walter [see Complex analysis and its applications, Collect. Artic., Steklov Math. Inst., Moscow 1978, 494-513 (1978; Zbl 0425.35053)]. Certain monotonicity assumptions are made with respect to the last two arguments of f and \(\phi\). Under the additional assumptions that \(Pu<Pv\) in Int D, \(u<v\) in \(\Sigma_ 0\cup \Sigma^*\), \(Ru<Rv\) in \(\Sigma\), it is proved that \(u<v\) in D. A similar theorem is proved with strong inequalities replaced by weak inequalities, and from this a uniqueness theorem is obtained. The method of the proofs is inspired by the work of J. Szarski [Ann. Polon. Math. 31, 197-203 (1975; Zbl 0315.35015)].
Reviewer: R.C.Gilbert

MSC:

35R45 Partial differential inequalities and systems of partial differential inequalities
35K20 Initial-boundary value problems for second-order parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
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