A Class Of Degenerate Parabolic Equations With A Concentrated Nonlinear Source

In this paper, we mainly consider the following Cauchy problem for the degenerate parabolic equation with a concentrated nonlinear sourcewhere m > 1, Q_t = R~n×(0,T), n≥2, B is an n-dimensionalball {x∈R~n : |x| < b}, (?)B is the boundary of B, xb(x) is thecharacteristic function of set B, v(x) denotes the unit inward normalat x∈(?)B.For the researce of the parabolic equations with a concentrated nonlinear source, mathematical workers have obtained some results. Chan and Tian considered the following first initial-boundary value problem where B (?)D, D is an n-dimensional bounded domain, n≥2, (?)D is the boundary of D, (?) denotes its closure. The authors converted this problem into a nonlinear integral equation and showed that the integral equation has a unique continuous solution u, which is a nondecreasingfunction with respect to t, then they proved that u is the unique solution of the problem (3)-(5). They also gave a criterion for u to blow up in a finite time, and if u blows up, it was shown that it blows up somewhere on B. Later, they gave a different criterion which make u blow up everywhere on dB in a finite time. We can see that for the problem of heat equation with this source, some good results have been achieved.For the case of equations without a concentrated source, namely the problem of the following equationwith initial-boundary value, some results have been obtained. For m = 1, Associate Professor Ding obtained the nonexistence theorems of the global solutions by using the Hopf's maximum principle, and gave the bound of blow-up time. Furthermore, Professor Yin etc extended the similar results to the case m > 1 by utilizing regularized methods, and showed the existence of local solutions.When the right hand of equtions contain a concentrated source, Boccardo and Gallouet discussed the following problem where a is a nonlinear function satisfying some coerciveness and monotonicityassumptions, f is a bounded measure. The authors proved the existence of solutions for the problem (7)-(9). Then Professor Yuan etc considered the following Cauchy problemwhereδ(x) is Dirac measure. The authors obtained the existence of generalized solutions for the problem (10)-(11) based on some a prior estimates.In the present paper, the main purpose is to show the existence of solutions for the problem (l)-(2). Since m > 1. the appearance of the diffusion term makes the theory of Green's function used inapplicable. While due to the appearance of the concentrated sourceδ(x), the Hopf's maximum principle could no longer be applied to derive the L~∞estimate on solutions. In addition, since the source is associated with the nonlinear function f, some results such as the local L~1 estimate of the solutions can not be obtained by the methods used to deal with the problem (10)-(11). In this paper, by utilizing the comparison principle and some iteration technologies, we obtain the local L~1 estimate.In the second section of this paper, we introduce some necessary preliminaries and present the main results. In the third section, we consider the approximate problem of the problem (l)-(2), and make some a prior estimates on the approximate solutions. In the fourth section, we show that the problem (l)-(2) admits a generalized solutionon...