Boundary value problem for a class of degenerate quasilinear parabolic equations with singularity
This paper concerns the radial solutions \(u= u(r, t)\) \((|x|= r)\) of the following initial and boundary value problem for the quasilinear parabolic equations with arbitrary degeneracy: \[ \begin{aligned} u_t=\Delta A(u)+ \text{div}(B(u)x)\quad &\text{in }B_1(0)\times (0,T),\\ u= 0\quad &\text{on }\partial B_1(0)\times (0,T),\\ u(r,0)= u_0(r)\quad &\text{for }0\leq r< 1,\end{aligned} \] where \(B_1(0)= \{x\in\mathbb{R}^N: |x|< 1\}\), \(A(s)= \int^s_0 a(\tau)\,d\tau\), \(B(s)= \int^s_0 b(\tau)\,d\tau\), \(a(s)\geq 0\), and \(u_0\) is a given radial function. The assumption \(a(s)\geq 0\) allows the set of degeneracy \(\{s: a(s)= 0\}\) to have interior points. The authors prove the existence and uniqueness of generalized solutions by using the approximate solution \(u_\varepsilon\) of the following regularized problem for \(\varepsilon> 0\): \[ \begin{aligned} (u_\varepsilon)_t= {\partial^2 A_\varepsilon(u_\varepsilon)\over \partial r^2}+ {(N- 1)\over r+\varepsilon} {\partial A_\varepsilon(u_\varepsilon)\over \partial r}+ (r+ \varepsilon) {\partial B(u_\varepsilon)\over\partial r}+ NB(u_\varepsilon)\quad &\text{in }(0,1)\times (0,T),\\ u_\varepsilon(1, t)= {\partial u_\varepsilon(0, t)\over\partial r}= 0\quad &\text{for }t\in (0,T),\\ u_\varepsilon(r, 0)= u_0(r)\quad &\text{for }r\in (0,1),\end{aligned} \] where \(A_\varepsilon(s)= \int^s_0 (a(\tau)+ \varepsilon)\,d\tau\).
scientific article; zbMATH DE number 2093827