Show description

2) admits at most one generalized solution in i 1 ( Q ^ ) D L°°(QT). 1, we may obtain the following comparison theorem for the boundary value problem. 1) satisfying the initial value condition and boundary value condition ui(x,0) = Uio(x), for x G fl ( i = l , 2 ) and Ui(x,t)\r = 9i{x,t) for(x,t)€T (i = 1,2), Existence and Uniqueness of Solutions: Higher Dimensional Case 31 where £lcRN is a smooth domain, T = <9fi x (0,T), fiT = fi x (0,T). 1) on any domain of the form fi^ = fi x (0,T), we can use the comparison theorem on any fix for generalized solutions of the Cauchy problem.

26) yields du™ x Jo JMN tu I ^ dt 2 I dxdt < C. 2). 5 is thus completed. *)IU-(R") ^ tN/^t-m^i^r1^ (3-2§) with a constant C independent of UQ. This estimate can also be obtained by using the regularization method [FR1]. A similar estimate holds for the boundary value problem [EV]. 28) expresses a kind of regularization effect. 4, in some cases, the initial data might be a measure. [PIE] has proved the existence of generalized solutions for the case that the initial value is a nonnegative finite Radon measure.

Z„ achieves its minimum min^ z„ on a; = n, 0 < t < T. 13) yields 0 < -M<0, dv„ dx o o a s s - > o o and A'(s) > 0 imply the existence of the inverse function A _1 (w„). 15) yields Gn), x + \At\ll2 G \-n,n\ with At = t-s. x+\At\1/2 L un(z,t) t = rx+\At\ -un{z,s))dz 1/2 ^dydT I.