2. Let us fix T > 0. Consider the square Q = {x € R² : 0 < x₁ < л, 0 < x₂ < л} and the linear evolution problem U₁Ux1x₁x

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2. Let us fix T > 0. Consider the square Q = {x € R² : 0 < x₁ < л, 0 < x₂ < л} and the linear evolution problem U₁Ux1x₁x₂x₂ = f, in (0, T) × Q u(0, x) = g(x) , for x € Ω (1) u(t, x) = 0 , for all t = [0, T] and x = 0, where g € L²(Q) and ƒ € L²((0, T), L²(Q)). a) Let us set B(u, v) = -S₁₂ Vu(x).Vv(x)dx, \u, v € H¹(Q). Show that u € L²([0, T], H¹(Q)) ^ H¹([0, T], H¹(N)*) (u' (t), 9) H¹*.¸H¹ + B(u(t), y) = (f(t), 9)2, \ € ¹(N), for a.e. t € (0, T) u(0) = g is a weak formulation of the linear evolution problem (1), where g in the last equation is identified through the embedding L²(Q) ℃ H¹(Q)* with an element of H¹(Q)*. [5 marks]