$L_1$ - $L_\infty$ оценки решения задачи Коши для анизотропного вырождающегося параболического уравнения с двойной нелинейностью и растущими начальными данными

917.956.45 We study the phenomenon of instantaneous shrinking of the support of solution to the Cauchy problem for the parabolic equation with anisotropic degeneration, double nonlinearity, and strong absorption. In terms of the behavior of locally integrable initial data, we formulate necessary and sufficient conditions for the realization of instantaneous shrinking and establish the exact (in order) bilateral estimates for the size of the support of solution. Statement of the Problem and Main Resultwhere N is the dimension of the space R N , T > 0, we consider the following Cauchy problem for the unknown function u(x, t):where β > 0, p i > 0, i = 1, N, and λ > 0 are given constants and |u 0 | β−1 u 0 (x) is a given initial locally integrable function. We consider the case of slow diffusion in all directions and strong adsorption, which imposes the following restrictions on the parameters of the problem:In the case of isotropic equation, i.e., for p i = p, i = 1, N, it is known that, under certain conditions imposed on the initial function |u 0 | β−1 u 0 (x), problem (1.1), (1.2) is solvable in a weak sense and the phenomenon of instantaneous shrinking of the support is observed (see [1][2][3][4][5][6][7][8][9][10]). This phenomenon can de described as follows: the support of the solution becomes compact at any arbitrarily small time t > 0 and shrinks for small t despite the fact that the support of the initial function coincides with entire R N . Here, we do not analyze the history of the problem in detail (see [1-10]) but note, in particular, that the following exact (in order) bilateral estimate for the size D(t) of the support of solution to problem (1.1), (1.2) was obtained for the case of isotropic equation in [10]:where D(t) = inf r : u(x, t) ≡ 0, |x| > r ,

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