A well posed problem in singular Fickian diffusion

Abstract

We prove that the problem of solving

$$u_t = (u^{m - 1} u_x )_x {\text{ for }} - 1< m \leqq 0$$

with initial conditionu(x, 0)=φ(x) and flux conditions at infinity$\mathop {\lim }\limits_{x \to \infty } u^{m - 1} u_x = - f(t),\mathop {\lim }\limits_{x \to - \infty } u^{m - 1} u_x = g(t)$, admits a unique solution$u \in C^\infty \{ - \infty< x< \infty ,0< t< T\}$ for every φεL¹(R), φ≧0, φ≡0 and every pair of nonnegative flux functionsf, g ε L ^∞_loc [0, ∞) The maximal existence time is given by

$$T = \sup \left\{ {t:\smallint \phi (x)dx > \int\limits_0^t {[f} (s) + g(s)]ds} \right\}$$

This mixed problem is ill posed for anym outside the above specified range.