Local structure of solutions of the reaction-diffusion equations

✍ Scribed by Ugur G. Abdullaev

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 538 KB
• Volume: 30
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(96)00283-0

✦ Synopsis

We consider the initial-value problem for the nonlinear parabolic equation with u, -a(u")\_ + bd = 0, -coo u(x,O) = t&(x). -w < x < m ) and a > 0, b E R', m 2 1, ,B > 0 The inital function has finite support and is supposed to be nonnegative, and continuous. Locating the right-hand edge of the support of y(x) at the point x = I, we assume also the initial function to be smooth in (I -6,f) , for some S > 0. We show that the small-time behaviour of the interface, which emerges from the point (x,t) = (I,O) , as well as the local structure of solution near the interface depend crucially on the number y = jjpo(u(u;)"/bu:)
In all possible cases, when interface either shrinks or remains stationary, the small-time behaviour of the interface is found, together with the local solution.