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Linear and nonlinear abstract equations with parameters. (English) Zbl 1215.34067
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.

34G10Linear ODE in abstract spaces
35J25Second order elliptic equations, boundary value problems
35J70Degenerate elliptic equations
34G20Nonlinear ODE in abstract spaces
34B10Nonlocal and multipoint boundary value problems for ODE
47D06One-parameter semigroups and linear evolution equations
Full Text: DOI
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