Parabolic maximum principles

For a bounded domain Ω and times 0 through T, the parabolic boundary is the initial slice together with the lateral spatial boundary over time. Let u be continuous on the closed cylinder and C² in space and C¹ in positive time in the interior. If uₜ−κΔu≤0 with κ>0, its supremum does not exceed the supremum on that parabolic boundary. No terminal boundary value is required to solve the forward problem, and the final time may be included among the times being bounded.

At a positive interior spatial maximum, the Hessian is negative semidefinite and the Laplacian is nonpositive. At a maximum reached at the final time, the left time derivative is nonnegative; one should not simply set it to zero. Subtracting a small positive multiple of time makes the subsolution inequality strict and rules out an interior maximum above the initial and lateral values. Letting the perturbation vanish yields the non-strict principle. This proof is for a scalar classical solution; applying it directly to an arbitrary weak representative would require a separate weak comparison argument.

For the scalar operator uₜ−aᵢⱼuᵢⱼ−bᵢuᵢ+cu with bounded continuous coefficients, symmetric uniformly positive definite a, and c≥0, the corresponding subsolution bound uses the maximum of zero and the parabolic-boundary supremum. At a positive maximum the drift term vanishes, the diffusion contribution has the correct sign, and cu is nonnegative. The zero term in the bound matters when all boundary values are negative. A negative zero-order coefficient can instead permit positive growth.

If a bounded coefficient gives uₜ−κΔu=a(x,t)u, choose L at least its upper bound and transform v=e⁻ᴸᵗu. The transformed equation has a nonnegative zero-order coefficient L−a in the left-side convention. This supplies a weighted comparison bound rather than an unweighted claim that every maximum decreases. For a scalar reaction F(u) locally Lipschitz on the bounded ranges of two solutions, their difference has such a bounded coefficient. An exponential weight therefore supports comparison without requiring F itself to be decreasing.

A supersolution lies above the PDE's forcing after applying the same operator and lies above the initial and boundary values. Subtracting it from a solution reduces the desired inequality to a maximum principle. Spatially constant time-dependent functions are often effective barriers when the forcing has a simple bound. For homogeneous heat flow, comparison also preserves nonnegativity. Strong positivity requires further hypotheses, such as a connected region and nontrivial nonnegative data, and should not be confused with the weak upper-bound statement.

Maximum principles establish bounds for solutions in a class where comparison is valid; they do not construct those solutions by themselves. They are naturally scalar and need not hold componentwise for an arbitrary coupled parabolic system. Boundary geometry and regularity also matter when moving from interior estimates to continuous attainment of boundary data. A reliable application states the operator sign, the domain and time interval, the solution regularity, and the ordering on the entire parabolic boundary before invoking a convenient constant or time-dependent barrier.