摘要
This paper is devoted to study the classification of self-similar solutions to the m ≥ 1,p,q > 0 and p + q > m. For m = 1, it is shown that the very singular self-similar solution exists if and only if nq + (n + 1)p < n + 2, and in case of existence, such solution is unique. For m > 1, it is shown that very singular self-similar solutions exist if and only if 1 < m < 2 and nq + (n + 1)p < 2 + mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.
This paper is devoted to study the classification of self-similar solutions to the quasilinear parabolic equations with nonlinear gradient termsu t = Δ(u m) - uq∣Δu∣ p withm ≥ 1,p,q > 0 andp +q >m. Form = 1, it is shown that the very singular self-similar solution exists if and only ifnq + (n +1)p <n + 2, and in case of existence, such solution is unique. Form > 1, it is shown that very singular self-similar solutions exist if and only if 1 <m < 2 andnq + (n + 1)p < 2 +mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.
基金
This work was supported by the National Natural Science Foundation of China(Grant No.19831060)
the"333"Project of Jiangsu Province.