A probabilisitc solution is given by MR1469575 a nonlinear diffusion $$ Y_t = Y_0 + \int_0^t u^{\frac{m-1}{2}}(s,Y_s)\; \mathrm{d}W_s , \qquad \mathrm{law}(Y_0) = u(0,\cdot) $$ then $$ \mathrm{law}(Y_t) = u(t,\cdot) . $$ This is true for a general class of nonlinear diffusion equations. The best references I've found are MR1775228 and MR2722788.
André Schlichting
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