Nonnegative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders
Björn E. J. Dahlberg, Carlos E. Kenig · Journal of the American Mathematical Society · 1988
In this paper we study nonnegative solutions U of a class of nonlinear evolutions au/at = dqJ(U) , whose best known example is the porous medium equation qJ(u) = urn, m > 1 (see [9] for a recent survey on these equations).We study our solutions u, subject to Dirichlet boundary conditions in D x (0, T) , where D is a bounded smooth domain in R n , i.e., ulaDX(O.T) == O.The corresponding theory for D = R n was studied by the authors in [3].We find that there is one special solution P(x, t) (the 'friendly giant') which tends to infinity as t 1 0, and that all other solutions u(x, t) are in one-to-one correspondence with suitable pairs of measures J.l on D and A. 00 aD.We also show that any solution on D x (0, T) extends to a solution in D x (0,00), and that the friendly giant P(x, t) governs the asymptotic behavior as t i +00.The nonlinearity is assumed to be continuous, increasing, with qJ(O) = 0, and also verifies the growth conditionsfor some constant a, 0 O.We will denote by ra the class of qJ 's which verify the above conditions, together with the normalizationWe say that u is a strong solution of the initial Dirichlet problem (IDP) for the equation au/at = dqJ(U) in D x (0,00) if u is a continuous, nonnegative function in D x (0,00), u == 0 on aD x (0,00), and for all smooth functions If/ on Dx[O,oo),whichvanishon aD x [r 1 ,r 2 ], r 1 >O,wehave j e ( [qJ(U)dlf/ + U ~If/] dx dt JDX[TI.T2) t = In u(x, r 2 )If/(x, r 2 ) dx -In u(x, r1)If/(x, r1)dx .