Stability Problem Of A One-dimensional Heat Equation In Non-cylindrical Domains

In this paper we mainly consider the following system of one-dimensional heat equation in non-cylindrical domains. hereFirst we take the one-dimensional homogeneous heat equation system with right-side moving boundary for an example, and we obtain Fourier type series solution for the prob-lem, and the solution is polynomial stability to zero. Then we consider the one-dimensional heat equation system we have mentioned.We use variable substitution to transform the heat equation of moving boundary into the parabolic equation with variable coefficient in the cylindrical region. Then, we use the energy method to prove if we give f some appropri-ate limits,and let a be constant or a= a?t?, t ? ?0,+??, c> 0, we have the following conclusions.?1?if b= 0,||u||L₂?D_t? is polynomial stability to zero.?2?if b> 0,||u||L₂?D_t? is exponential stability to zero.?3?if b= b?t? satisfies b?t?> c₀, c₀> 0,||u||L₂?D_t? is exponential stability to zero.That is to say, if we have different coefficients,then we can give heat source some appro-priate limits to make the solution exponential stability or polynomial stability. Furthermore, we describe the moving boundary and extend the conclusions, and we give examples to prove that when we choose different coefficients in non-cylindrical domains, the solutions are sta-bility to zero in different speed.