On Existence And Nonexistence Of The Positive Solutions Of Non-Newtonian Filtration Equation

Abstract

The subject this investigation is existence and nonexistence of positive solutions of the following nonhomogeneous equation rho(vertical bar x vertical bar) partial derivative u/partial derivative t - Sigma(N)(i=1) D(i)(u(m-1)vertical bar D(i)u vertical bar(lambda-1)D(i)u) + g (u) + lu = f (x) (1) or, after the change v = u(sigma), sigma = m+lambda-1/lambda, of equation rho(vertical bar x vertical bar) partial derivative v(1/sigma)/partial derivative t - sigma(-lambda) Sigma(N)(i=1)D(i)(vertical bar Div vertical bar(lambda-1)D(i)v) + g(v(1/sigma)) + lv(1/sigma) = f (x), (1') in unbounded domain R(+) x R(N), where the term g (s) is supposed to satisfy just a lower polynomial growth condition and g' (s) > -l(1). The existence of the solution in L(1+ 1/sigma)(0, T; L(1+ 1/sigma)(R(N))) boolean AND L(lambda+1)(0, T; W(1,lambda+1) (R(N))) is proved. Also, under some condition on g(s) and u(0) is shown a nonexistence of the solution.