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Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia. (Ukrainian, English) Zbl 1114.35112
Ukr. Mat. Zh. 58, No. 9, 1192-1210 (2006); translation in Ukr. Math. J. 58, No. 9, 1347-1368 (2006).
The authors prove an existence and uniqueness theorem for solutions of a mixed problem for the equation of the form \[ u_t+\sum \lambda_i(x,y,t)u_{y_i}-\sum (a_j(x,y,t)| u_{x_j}| ^{p- 2}u_{x_j})_{x_j}+c(x,y,t)u+g(x,y,t,u)=f(x,y,t) \] with the time variable \(t\) and two groups \(x_j,y_i\) of spatial variables. See S. D. Èidelman, S. D. Ivasyshen and A. N. Kochubei [Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. Basel: Birkhäuser (2004; Zbl 1062.35003)] for an exposition of the theory of linear equations of this type.

35K70 Ultraparabolic equations, pseudoparabolic equations, etc.
35K20 Initial-boundary value problems for second-order parabolic equations
Zbl 1062.35003
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