New blow-up conditions to p-Laplace type nonlinear parabolic equations under nonlinear boundary conditions

In this paper, we study blow-up phenomena of the following p-Laplace type nonlinear parabolic equations u(t) = del center dot rho(vertical bar del u vertical bar(p))vertical bar del u vertical bar(p-2) del u+f(x,t,u), in Omega x(0,t*), under nonlinear mixed boundary conditions rho(vertical bar del u vertical bar(p))vertical bar del u vertical bar(p-2)partial derivative u/partial derivative n+theta(z)rho(vertical bar u vertical bar vertical bar p)vertical bar u vertical bar p-2u=h(z,t,u),on Gamma 1x(0,t*), and u=0 on Gamma(2) x (0, t*) such that Gamma(1)boolean OR Gamma(2)= partial derivative omega, where f and h are real-valued C-1-functions. To discuss blow-up solutions, we introduce new conditions: For each x is an element of omega, z is an element of partial derivative Omega, t > 0, u > 0, and v > 0, (D(p)1):alpha F(x,t,u)<= uf(x,t,u)+beta 1up+gamma 1, alpha H(z,t,u)<= uh(z,t,u)+beta 2up+gamma 2, (Dp2):delta v rho(v)<= P(v), for some constants alpha, beta(1), beta(2), gamma(1), gamma(2), and delta satisfying alpha>2,delta>0,beta(1)+lambda(R)+1/lambda(S) beta(2)<= (alpha delta/p-1)rho(m)lambda(R), and 0 <=beta(2)<= (alpha delta/p-1)rho(m)lambda(S), where rho m:=infw>0 rho(w), P(v)=integral 0v rho(w)dw, F(x,t,u)=integral 0uf(x,t,w)dw, and (x,t,u)=integral 0uh(x,t,w)dw. Here, lambda(R) is the first Robin eigenvalue and lambda(S) is the first Steklov eigenvalue for the p-Laplace operator, respectively.