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Zbl 1215.34067
Shakhmurov, Veli
Linear and nonlinear abstract equations with parameters. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 8, 2383-2397 (2010). ISSN 0362-546X

The linear abstract equation $$-tu^{(2)}(x)+ Au(x)+ t^{1/2} B_1(x)u^{(1)}(x)+ B_2(x) u(x)= f(x)$$ with a parameter $t$ is considered. Here, $A$ and $B_1(x)$, $B_2(x)$ for $x\in (0,1)$ are linear operators in a Banach space. The nonlocal boundary conditions contain the parameter $t$ as well. Under some assumptions, the existence of the unique solution in a Sobolev space and a coercive uniform estimation is established. Also, the behavior of the solution for $t\to 0$ and the smoothness properties of the solution with respect to the parameter $t$ are investigated and the discreteness of the corresponding differential operator is proved. For the nonlinear problem with right side $f(x,u, u^{(1)})$, the existence and uniqueness of maximal regular solution is obtained. An application to the equation $$-t_1 D^2_x u(x,y)- t_2 D^2_y u(x,y)+ du(x,y)+ t^{1/2}_1 D_x u(x,y)+ t^{1/2}_2 D_y u(x,y)= f(x,y)$$ on the region $(0,a)\times (0,b)$ is given.
[S. Burys (Kraków)]

MSC 2000:: *34G10 Linear ODE in abstract spaces
35J25 Second order elliptic equations, boundary value problems
35J70 Elliptic equations of degenerate type
34G20 Nonlinear ODE in abstract spaces
34B10 Multipoint boundary value problems
47D06 One-parameter semigroups and linear evolution equations

Keywords: abstract boundary value problems; nonlocal conditions; equations with parameter

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