Quenching of solutions of semilinear Euler-Poisson-Darboux equations (Q2724360)

The authors consider the problem \(u_{tt}+(k/t)u_t-\triangle u=f(u)\) for \((x,t)\in D\times (0,T)\) with initial data \(u(x,0)=u_0(x)\), \(u_t(x,0)=0\) (\(x\in D\)) under the boundary condition \(u(x,t)=0\), \((x,t)\in \partial D\times [0,T)\), where \(D\) is a bounded \(n\)-dimensional domain with a piecewise smooth boundary \(\partial D\), \(k\) is any real number, \(\triangle \) is the \(n\)-dimensional Laplace operator and \(\lim_{u\to c^-}f(u)=\infty \) (\(c>0\)). Criteria for a weak solution \(u\) of the problem to reach the value \(c\) somewhere is established. The blow-up of the derivative \(u_t\) is discussed. It is proved that a weak solution of the considered problem must quench in a finite time for \(k\geq 1\) provided that \(f(s)-\lambda s \geq 0\) in \((0,c)\) and \(f(w(0))-\lambda w(0)>0\). Another result is that under the same requirements and if \(\int f(u)du=\infty \) then \(u_t\) blows up in a finite time for \(k\leq 1\).