Existence of positive solutions to a nonlinear initial problem (Q5940179)

The authors prove the existence of a positive decreasing solution \(f=f(y)\) of the problem \(f''=yf'/2+f/(p-1)-f^p\) for \(y>0\), \(f'(0)=-f^{(p+1)/2}(0)\). This problem plays an important role in the study of the blow-up rate of blowing up solutions of the parabolic equation \(u_t=u_{xx}+u^p\) complemented by the nonlinear boundary condition \(u_\nu=u^{(p+1)/2}\) (where \(u_\nu\) means the derivative with respect to the outer unit normal \(\nu\)).