A comparison principle for the porous media equation with absorption
The authors study the following initial boundary value problem \[ u_ t=\Delta(u^ m)-\lambda u^ p,\quad u\geq 0\quad\text{in }\Omega\times\mathbb{R}^ +,\quad u(x,t)=0\quad\text{on }\partial\Omega\times\mathbb{R}^ +,\quad u(x,0)=u_ 0(x)\quad\text{in }\Omega, \] where \(m\), \(p>0\) and \(\lambda\geq 0\) are constants, \(u_ 0\) is a given bounded nonnegative function and \(\Omega\) is a bounded connected open set in \(\mathbb{R}^ n\) whose boundary is of class \(C^ 3\). A supersolution of the problem is defined as usually to be a nonnegative bounded function \(\overline u\in C(\mathbb{R}^ + :L^ 1(\Omega))\) such that \[ \int u(T)\xi(T)\geq\int u_ 0\xi(0)+\int\int(u\xi_ t+u^ m\Delta\xi- \lambda u^ p\xi),\quad\forall T>0 \] for any nonnegative test function \(\xi\) belonging to \(C^ 2(\overline\Omega\times[0,T])\) vanishing on \(\partial\Omega\times(0,+\infty)\), (Subsolutions \(\underline u\) are defined similarly, with \(\geq\) replaced by \(\leq)\). The aim of the paper is to prove a comparison principle i.e. to show that if \(\overline u(x,0)\geq\underline u(x,0)\) then \(\overline u(x,t)\geq\underline u(x,t)\). Moreover, if \(m\in(0,1)\) and \(p\geq m\), then there exists an extinction time \(T^*>0\) such that \(u=0\) in \(\overline\Omega\times(T^*,+\infty)\) and \(u>0\) in \(\Omega\times(0,T^*)\), once the initial case \(u_ 0\equiv 0\) has been excluded.
scientific article; zbMATH DE number 60458
scientific article