Existence of classical solutions for degenerate semilinear parabolic problems (Q1294245)

The article is devoted to studying solvability and properties of solutions to the following initial-boundary value problem: \[ x^qu_t = u_{xx} + f(u)\qquad\text{in } \Omega_T = (0,1)\times (0,T], \] \[ u(0,t) = u(1,t) = 0\qquad\text{for } 0<t\leq T, \] \[ u(x,0) = u_0(x)\geq 0\qquad\text{on } [0,1], \] where \(f(0) = 0\), \(f'(0)\geq 0\), and \(f''(u) > 0\), and \(u_0\in C^{2+\alpha} ([0,1])\) for some constant \(\alpha\in (0,1)\). The authors construct Green's function corresponding to the problem, and establish the existence of a unique classical solution of its corresponding linear problem through the representation formula. Using the monotone method, it is established the existence of a unique classical solution \(u\) of the problem through successive approximations by a sequence of linear problems. The equivalent integral equation for \(u\) is demonstrated. Such a formulation is useful for studying the behavior of the solution \(u\), such as the complete blow-up for example, and it is shown that the solution either exists globally or blows up in finite time.