On the influence of the growth of boundary data on the behavior of the temperature of a nonlinear nonstationary medium for large time values (Q1819287)

The paper studies some properties of the generalized solutions of the partial differential equation \[ [a(x,t)(u^ m)_ x]_ x+[b(x,t)u^ n]_ x-c(x,t)u^ p-u_ t=0, \] where m, n, p are constants (2\(\leq m\leq n,p)\), a(x,t), \(a_ x(x,t)\), b(x,t), \(b_ x(x,t)\), c(x,t) are positive definite and continuous on \({\underline {\mathbb{R}}}^ 2_+\) and bounded for bounded t. There are established many conditions for the existence and the boundedness of the generalized solutions and the influence of the growth of boundary data on the behaviour of the temperature of a nonlinear non-stationary medium at large values of the time.