On the fundamental solution of the operator of dynamic linear thermodiffusion
This paper deals with the construction of the fundamental matrix to the linear system of thermodiffusion \[ A(\partial)(u,T,p)^T= (F,Q,M)^T,\tag{\(*\)} \] where \(u\) is the vector field of displacement, \(T\), \(p\) are the scalar fields of temperature and chemical potential, respectively, and \((F,Q,M)\) describes outer influences. \(A\) is a \(5\times5\) matrix of appropriate partial derivatives. It was shown by the authors in earlier papers that if \(E_W\) denotes a fundamental solution of the iterated wave operator \(W(\partial)= \rho(1-\sigma_1\sigma_2) (\rho\partial^2_t- \mu\Delta)^2\), which can be given explicitly, then the fundamental solution of \((*)\) \(E_A\) is given as \(E_A=\text{adjoint} (A(\partial))E_W* E_p\), where \(E_p\) is the fundamental solution of the ``operator of dynamic linear thermodiffusion'' \[ \partial^4_t+a\partial^3_t \Delta+b\partial^2_t\Delta^2+ c\partial^2_t\Delta+ d\partial_t\Delta^2+e\Delta^3 \] such that \(e^{-\sigma t}E_p\) is a temperature distribution for sufficiently large \(\sigma\). The paper is concerned with the construction of the latter fundamental solution \(E_p\) by integral representations.
scientific article; zbMATH DE number 856810