A Phragmen-Lindelöf alternative for a class of quasilinear second order parabolic problems
The authors consider classical solutions of the quasilinear equation \((\rho(x^j, t, u, \nabla u)_{,j})_{,j}= u_t\), \(i,j= 1,2,3\) in a semi-infinite cylinder in \(\mathbb{R}^3\) of cross section \(D\). The function \(\rho\) satisfies either \(0< m_1\leq \rho\leq [M_1+ K_1 \rho q^2]^{1/2}\), \(q^2= |\nabla u|^2\) or \(0< m_2\leq {1/\rho}\), where \(M_1, K_1> 0\). Then they show that the function \[ \Phi(z, t)= \int^t_0 \int_{R_z} \rho_{,j} u_{,j} dx d\tau+ \textstyle{{1\over 2}}\displaystyle \int_{R_z} u^2 dx, \] where \(R_z= \{(x_1, x_2, x_3): (x_1, x_2) \in D,\;x_3> z> 0\}\), becomes unbounded or decays to zero as \(z\to \infty\). Here \(u\) satisfies zero Dirichlet boundary conditions on \(\partial D\), zero initial value and \(u(x_1, x_2, 0, t)\) is prescribed.
quasilinear second-order parabolic problems
scientific article; zbMATH DE number 716368