Integrodifferential reformulation of the Cauchy problem for a quasilinear heat conduction equation

We consider the Cauchy problem for the quasilinear heat conduction equation with a variable heat capacity coefficient and a heat transfer coefficient proportional to temperature. The Cauchy problem for the original equation is reduced to a dual problem for some integrodifferential equation for the Fourier image of the desired solution with initial data on the positive semi-axis. Integration in the obtained equation for the Fourier image of the solution of the initial differential problem is performed over the first quadrant of the plane of independent variables. The bilinear integral operator in the obtained integrodifferential equation has as a kernel a function of time and two non-negative integration variables. The kernel is explicitly expressed through the variable heat capacity coefficient of the original differential equation.