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Vanishing of solutions of diffusion equation with convection and absorption. (English) Zbl 1090.35108
The authors consider the Cauchy problem for the equation
\[ u_t= \sum^N_{i,j=1} a_{ij}(u^m)_{x_ix_j}+ \sum^N_{i=1} b_i(u^n)_{x_i}- cu^p,\;(x,t)\in \mathbb{R}^N\times (0,\infty)\tag{1} \] with the initial data \(u(x,0)= u_0(x)\), \(x\in\mathbb{R}^N\), where \(m> 1> p> 0\), \(n\geq 1\), \(a_{ij}\), \(b_i\) \((i,j= 1,\dots, N)\), \(c\) are real numbers, \(a_{ij}= a_{ji}\), \(\sum^N_{i,j=1} a_{ij}\xi_i\xi_j> 0\) for \(\sum^N_{i=1} \xi^2_i> 0\) (\(\xi_i\in\mathbb{R}\), \(i= 1,\dots, N\)), \(c> 0\), \(u_0(x)\) is a nonnegative continuous function which can be increasing at infinity. Equation (1), due to the degeneracy, can have nonclassical solutions even when initial data are smooth. The authors present existence and uniqueness of solutions.

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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