An inverse blow-up problem (Q324117)

The authors consider the problem of finding the function \(f(u)\) associated with the second order nonlinear equation NEWLINE\[NEWLINE y''(x)=f(y(x)),\; y(0)=h \; \text{and}\;\;y'(0)=0NEWLINE\]NEWLINE from the knowledge of the blow-up time function given by NEWLINE\[NEWLINET_f(h)=\frac{1}{\sqrt{2}}\int_h^{\infty}\frac{du}{\sqrt{\int_h^uf(\xi)d\xi}}.NEWLINE\]NEWLINE The first result deals with the global continuation of the function \(f\) when \(T_f\) is positive and continuous. The second result shows local existence and a uniqueness theorem when \(T_f\) has some known asymptotics.