I want to prove the existence and uniqueness of the following problem by the semi-group method $$ \eqalign{ & {u_{tt}} + {u_{xxtt}} + {u_t} - {u_{xx}} = 0 \cr & u(t,0) = u(t,l) = 0 \cr & u(0) = {u_{0{\rm{ }}}} ,u'(0) = {u_1} \cr} $$ I wrote it as a Cauchy problem : $$U' = AU$$ with : $U = ({v^1},{v^2})$ and $A$ is defined as : $$ A = \begin{pmatrix} 0 & I \\[3pt] \Big(I + \frac{\partial ^2}{\partial x^2}\Big)^{ - 1}\!\!\frac{\partial ^2}{\partial x^2} & - \Big(I + \frac{\partial ^2}{\partial x^2}\Big)^{ - 1} \end{pmatrix} $$ My question is how can I treat the operator ${(I + {{{\partial ^2}} \over {\partial {x^2}}})^{ - 1}}$ ? Thank you
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What about Fourier series? – guacho Jan 16 '17 at 09:49
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@guacho: Not good on a bounded domain... – DominikS Aug 10 '17 at 15:00
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Is this post related to your issue? https://math.stackexchange.com/questions/32932/how-can-one-relate-inverse-of-a-differential-operator-to-an-integral-operator – Jan 03 '18 at 09:02