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Eindeutigkeitsaussagen für das Tricomi-Problem im \({\mathbb{R}}^ 2\) für eine Klasse nichtlinearer Gleichungen gemischten Typs. (German) Zbl 0532.35055

For the nonlinear equation \(L[u]:=k(y)u_{xx}+u_{yy}=g(x,y,u,u_ x,u_ y)\quad(sign k(y)=sign y)\) in G, bounded by a piecewise smooth curve \(\Gamma_ 0\) for \(y>0\) which intersects the line \(y=0\) at the points \(A(-1,0)\) and B(0,0) and for \(y<0\) by the characteristics \(\Gamma_ 1\) and \(\Gamma_ 2\) through the points A and B which intersect at the point C the uniqueness of boundary value problems \(i)\quad u|_{\Gamma_ 0\cup \Gamma_ 1}=\phi\) and \(ii)\quad d_ nu|_{\Gamma_ 0}=k(y)u_ xd_ y-u_ ydx=\phi ds\), \(u|_{\Gamma_ 1}=\psi\) is proved by the energy-integral method. To prove uniqueness theorems a growth condition for the function \(g_ u(...)\) in the hyperbolic part of the region is needed.

MSC:

35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35G20 Nonlinear higher-order PDEs
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References:

[1] Aziz, A. K., Schneider, M.: On uniqueness of Frankl-Morawetz problem in ?2. Mh. Math.85, 265-276 (1977). · Zbl 0397.35048 · doi:10.1007/BF01305956
[2] Aziz, A. K., Schneider, M.: The existence of generalized solutions for a class of quasilinear equations of mixed type. J. Math. anal. appl. (To appear.) · Zbl 0592.35094
[3] Gvazava, D. K.: On uniqueness of solutions of the tricomi problem for a class of nonlinear equations. Soviet Math. Dokl.11,65-69 (1970). · Zbl 0198.43703
[4] Gvazava, D. K.: Some estimates of the magnitude of solutions of nonlinear equations of mixed type. J. Diff. Equat.6, 40-44 (1972). · Zbl 0247.35086
[5] Gvazava, D. K.: On some Classes of quasilinear Equations of Mixed Type. (In Russian.) Tr. Tbilis. Mat. Inst. Razmadze 67. 1981. · Zbl 0481.35068
[6] Saigo, M.: On uniqueness and estimations of solutions of modified Frankl problem for linear and nonlinear equations of mixed type. Proc. Japan Acad.48, 28-33 (1972). · Zbl 0238.35058 · doi:10.3792/pja/1195519807
[7] Saigo, M.: On the solutions of the modified Frankl problem. Kumamoto J. Sci. (Math.)9, 41-54 (1972). · Zbl 0238.35059
[8] Stoll, R. R.: Linear Algebra and Matrix Theory. New York: MacGraw-Hill. 1952. · Zbl 0049.29404
[9] Schneider, M.: Funktionalanalytische Methoden zur Behandlung von linearen partiellen Differentialgleichungen vom gemischten Typ. In: ?berblicke Mathematik, Bd. 7, pp. 219-265. Mannheim-Z?rich: Bibliographisches Inst. 1974.
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