Topological structure of solution sets to parabolic problems

In this paper we deal with the Peano phenomenon for general initial-boundary value problems of quasilinear parabolic equations with arbitrary even order space derivatives. The nonlinearity is assumed to be a continuous or continuously Frechet differentiable function. Using a method of transformation to an operator equation and employing the theory of proper, Fredholm (linear and nonlinear) and Nemitskii operators, we study the existence of solution of the given problem and qualitative and quantitative structure of its solution and bifurcation sets. These results can be applied to the different technical and natural science models.