Finite extinction time for some perturbed Hamilton-Jacobi equations
A class of Hamilton-Jacobi like equations \(u_ t-R|\nabla u|^ m+\lambda u^ q=0\) are considered with initial condition \(u_ 0(\cdot)=u(\cdot,0)\) satisfying \(-R|\nabla u_ 0|^ m+\lambda u_ 0^ q\geq 0\). It is shown that these equations have minimum non- negative viscosity solutions which are zero after some time \(t_ \infty(x)\), i.e. \(u(x,t)>0\) if \(0<t<t_ \infty(x)\), \(u(x,t)=0\) if \(t\geq t_ \infty(x)\). Differentiability of \(u\) at the support boundary is shown and the Lipschitz continuity of \(t_ \infty(x)\) is also proved. The extinction rate of \(u\) is considered in detail and under certain conditions it is shown that the exact value can be obtained. Finally the representation formula \(u(x,t)=\max\{\gamma^{-1}(\gamma(u_ 0(x-\xi t))-\lambda t):\;|\xi|\leq R\}\) is obtained.
minimum non-negative viscosity solutions
scientific article; zbMATH DE number 147398
scientific article