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Exact solution of a degenerate fully nonlinear diffusion equation. (English) Zbl 1058.35122

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 [P. Broadbridge, J. M. Goard and 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 \(y_t=F(u_x)\).

MSC:

35K65 Degenerate parabolic equations
35C05 Solutions to PDEs in closed form
35K57 Reaction-diffusion equations
76R50 Diffusion
35K15 Initial value problems for second-order parabolic equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs

Citations:

Zbl 0893.76086
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