Exact solution of a degenerate fully nonlinear diffusion equation
Nonlinear parabolic equations of the form \[ y_t=F(y_{xx})=\frac{\alpha^2}{4\beta}\big(1+\exp(-2\beta y_{xx})- 2\exp(-\beta y_{xx})\big),\tag{1} \] with \(\alpha\) and \(\beta\) positive constants, are considered. Using the definition \(u=y_x\) the equation (1) is reduced to equation \(u_t= \frac{\alpha^2}2 (\exp(-\beta u_{x})-\exp(-2\beta u_{x}))u_{xx}\). By a minor modification of the approach used in [\textit{P. Broadbridge, J. M. Goard} and \textit{M. Lavrentiev, jun.} Stud. Appl. Math. 99, No. 4, 377--391 (1997; Zbl 0893.76086)] a generalized solution: \(u=x/| x| \) for \(| x| >\alpha\sqrt{t}\), \(u=\frac{u}{\beta} \left(1- \log(| x| / \alpha\sqrt{t})+(m-\alpha\sqrt{t}/\beta) x/| x| \right)\) for \(0<| x| <\alpha\sqrt{t}\), and \(u(0,t)=0\), satisfying initial condition \(u(x,0)= m\text{sign}(x)\) is constructed. The solution to (1) is obtained by integrating the equation \(y_t=F(u_x)\).
scientific article; zbMATH DE number 2123178