A nonexistence result for a nonlinear PDE with Robin condition - Brahim Khodja (International Journal of Mathematics and Mathematical Sciences)
Under the assumption λ>0 and f verifying f(x,y,0)=0 in D, 2F(x,y,u)−uf(x,y,u)≥0, u≠0, and if Ω=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)=∫0sf(x,y,t)dt).