On the existence, uniqueness and the stability of a solution to a cooling problem, for an isotropic 3-D solid

We consider the following cooling problem for an isotropic 3-D body:cΩρΩ∂tu(x,t)=κΩΔu(x,t)−β(u(x,t)−ub)α;β>0;u∈H2(Ω),x∈Ω,t>0;ub⩽u(x,t)⩽ua;1⩽α⩽3;subject to:γ1u=0(conditionfornoexternalheatenergysourceonΓ)cΓρΓ∂tγ0u(y,t)=κΓΔsγ0u(y,t)−kγ0u(y,t)−ubα;y∈Γ,where, t, time; ua, initial temperature state for the body (assumed constant); u, solid absolute temperature; γ0u, solid absolute surface temperature; γ1u, ∇u.n=∂u∂n; where n is the unit normal to the surface Γ; ub, temperature of the environment surrounding the body (assumed constant); k, radiation term constant; Δ, ∇s·∇=∇2; Δs, ∇s·∇s; where ∇s represents surface gradient; ∇:=∇s+γ1n; Hm(Ω), Sobolev space on the open bounded domain Ω; in our case m=1 or 2; Lq(Ω), Lebesgue space on the open bounded domain Ω; in our case q=2; ∂t, the operator ∂∂t.