A well posed problem in singular Fickian diffusion

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 $$T = sup left{ {t:smallint phi (x)dx > intlimits_0^t {[f} (s) + g(s)]ds} right}$$ This mixed problem is ill posed for anym outside the above specified range.