Introduction to Semigroups for Evolution Equations

u ′t + Aut = ft, t ∈ 0,T u0 = u0, where Au,vH = Bu,v ∀u,v ∈ V ⊂ H ⊂ V ′. We showed that if the bilinear form B is coercive then the linear mapping A is an isomorphism from V onto V’. Recall that there exists a family of H-orthonormal eigenfunctions for A such that ||Awk ||H = |λk| ||wk ||H = |λk|→ ∞ as k → ∞. Then A is clearly not bounded as a mapping from H into H. If we define DA = u ∈ H : Au ∈ H then H0U ∩ H2U ⊂ DA and A : DA → H Here we are saying that A restricted to DA takes its values in H rather than V’ but A is still not bounded in the norm of H. Note, however, that if un ⊂ DA, un → u in H, and Aun → v in H then un → u in D′U, and Aun → v in D′U. This means v = Au in the sense of distributions which implies in turn that Au,φ = v,φ ∀φ ∈ DU and since the test functions are dense in H, it follows that v = Au in H, so u ∈ DA and Au = v. Any operator with the property that un ⊂ DA, un → u in H, and Aun → v in H implies u ∈ DA and Au = v