A nonexistence result for a nonlinear PDE with Robin condition

Under the assumption λ > 0 and f verifying f (x, y,0)= 0 in D, 2F(x, y,u)−u f (x, y,u)≥ 0, u = 0, and ifΩ= R×D, we show the convexity of function E(t)= ∫∫D |u(t,x, y)|2dxdy, where u :Ω→R is a solution of problem λ(∂2u/∂t2)−(∂/∂x)(p(x, y)(∂u/∂x))−(∂/∂y)(q(x, y)(∂u/∂y)) + f (x, y,u)= 0 in Ω, u+ ε(∂u/∂n)= 0 on ∂Ω, considered in H2(Ω)∩L∞(Ω), p,q :D→R are two nonnull functions on D, ε is a positive real number, and D = ]a1,b1[ ×]a2,b2[, (F(x, y,s)= ∫ s 0 f (x, y, t)dt).