The initial trace of a solution of the porous medium equation

Let u = u(x, t) be a continuous weak solution of the porous medium equation in Rd X (0, TI for some T > 0. We show that corresponding to u there is a unique nonnegative Borel measure p on Rd which is the initial trace of u. Moreover, we show that the initial trace p must belong to a certain growth class. Roughly speaking, this growth restriction shows that there are no solutions of the porous medium equation whose pressure grows, on average, more rapidly then Ix 2 as Ix ) 00. Introduction. Let u = u(x, t) be a nonnegative solution of the equation of heat conduction (1) ~~~~~~au/at = Au in ST Rd X (0, T] for some T > 0. A consequence of the Widder representation theorem is the existence of a unique nonnegative Borel measure p such that (2) lim (x, t)(p(x) dx = (x)p(dx) for all test functions q E CO(Rd) [1]. That is, every nonnegative solution u of (1) has a unique Borel measure p as initial trace in the sense of (2). The measure p is a-finite and satisfies the growth condition (3) Lde-2/4Tp(d{) 1 is a constant. In addition, we show that the initial trace p must satisfy a growth condition analogous to (3) which limits the amount of mass which p can place at I x I = + x . Specifically, there exists a constant C > 0 depending only on d and m such that (5) f p(dx) ? C{rK/(m-l)T-l/(m-l) + Td/2UK/2 (0, T)} {x: IxI 0 and let g(x, t) = (47t)-d/2e_lX12/41 It is known [1] that the function u(x, t) _f g(x (, t)p(d() is a nonnegative solution of (1) in ST whose initial trace is p. Recently Benilan, Crandall and Pierre [4] have constructed solutions of the porous medium equation (4) whose initial traces are measures which satisfy (5). This extends some earlier work by Kalashnikov [7] for measures with smooth densities on R. Moreover, it shows that the condition (5) is not only necessary but also sufficient for existence of solutions of (4). Even more recently Dahlberg and Kenig [9] have proved uniqueness for the solutions constructed in [4]. 1. Comparison principles. Throughout this paper we will be dealing with continuous weak solutions of the porous medium equation (1.1) au/at =A (Um) in ST XRd X (0, T] for some T > 0. We always assume that m > 1. Specifically, a function u = u(x, t) is said to be a continuous weak solution of (1.1) in ST if u is nonnegative and continuous in ST and satisfies the integral identity (1.2) ffAqRdx (um p+ U-at) dx dt =| uP dx-f ucp dx for all Ti such that 0 0, and v satisfies the integral identity (1.5) ff(vlA9 + v a) dxdt Tfj gnl aLids dt +fv p dx-fgcp dx Br aIB t=T IB t=T for all p E C' 0(F) n C2'1(F) with 9p = 0 on aBr X [Tr, T2]. In (1.5), a/av denotes the exterior normal derivative on aBr. There is an a priori comparison principle for solutions of (1.4). This is the content of the next result, which is simply a restatement of Theorem 12 of [3]. PROPOSITION 1. 1 (LOCAL COMPARISON PRINCIPLE). For i = 1 and 2, let v, denote a solution of (1.4) with boundary values g,. If g1 g on ap F, then v in F. Proposition 1.1 is proved in [3] for x E R, but there is no difficulty in extending the proof for x E Rd. A proof for x E Rd is also given in [9]. The local comparison principle, Proposition 1.1, can be applied to continuous weak solutions of (1.1) since, as we show next, every such solution is also a solution of (1.4) with boundary values g = ulapr. PROPOSITION 1.2. Let u be a continuous weak solution of (1.1) is ST for some T > 0. For any cylinder r of the form (1.3), u is' a weak solution of (1.4) in F with boundary values g =uap. PROOF. It suffices to show that the weak solutions of (1.1) satisfy (1.5). Fix t E Rd, r E R+, 0 C T 2, (d 2)(r q)d-2(rE)d-2 We d-2 d-2 (r ) _-(re)d As a distribution on Rd, A4 is a signed measure. Specifically, (A4? n)(dx) WfJ Sr-q(dI x 1)r-3(dI X I)} dw, where d, denotes the surface element on the unit ball in Rd and 8x(d I x 1) denotes the Dirac measure concentrated at I x I= A. Similar expressions can be derived for d = I and 2. Let ky(x) be a C' symmetric averaging kernel with support in BY(O) and define 4Q,(x ky(x )4e( ) dt.