On a class of \(L^1\)-illposed quasilinear parabolic equations (Q1575817)

The authors study the Cauchy problem for the porous medium type parabolic differential equation \(u_t =\psi (u)\) in \(\mathbb R^N\times (0,T)\), where \(\psi\) is a smooth, increasing function on \((0,\infty)\) which is uniformly bounded from above and \(N \geq 2\). In addition, \(\psi\) is assumed to satisfy the conditions \(\limsup_{u\to\infty} \psi'(u)<\infty\) and \(-\psi(u) \geq C_0u^{1-n}\) for \(0<u<\delta\), where \(C_0\), \(\delta\), and \(n\) are positive constants. The model function \(\psi\) is \(\psi(u)=u^m/m\) for some negative number \(m\) (or \(\psi(u)=\log u\)). Under these assumptions, the authors show that any nonnegative solution of this equation must have initial data \(u_0\) which are not in \(L^1(\mathbb R^N)\). More specifically, they estimate the asymptotic behavior of the \(L^1\) norm of \(u_0\) over a ball of radius \(R\) in terms of \(R\), \(T\), \(N\), \(n\), and \(C_0\).