Blow-up criteria and continuation

Consider uₜ=κΔu+uᵖ on a connected periodic torus, with κ>0, p>1, and nonnegative continuous bounded initial data. The power is interpreted on nonnegative states, a class preserved by scalar comparison. The heat semigroup on continuous periodic functions and local Lipschitz control of the reaction on bounded ranges give a unique local mild solution. It is classical at positive times with the regularity supported by the reaction. Let Tmax be the supremum of times to which this bounded-solution construction extends.

If Tmax is finite, the supremum norm tends to infinity as time approaches it from below. The reason is that a uniform amplitude bound gives a uniform local Lipschitz constant for the reaction and a uniform additional existence interval, permitting a restart before Tmax and extension beyond it. This is a continuation criterion for this scalar semilinear framework. A different equation, such as a quasilinear diffusion that can lose ellipticity, may require a stronger norm or a separate lower-positivity bound. One cannot transfer the same criterion to every parabolic problem.

For constant initial data a>0, the Laplacian remains zero and the PDE reduces exactly to y′=yᵖ. Its solution blows up at a finite explicit time. Diffusion cannot suppress the reaction on this branch because there is no spatial variation for it to dissipate. This example proves that smoothing and global existence are distinct properties. It also gives comparison barriers: a maximum initial value supplies an upper ODE solution and a positive minimum supplies a lower one for as long as the relevant solutions remain finite.

An upper ODE barrier yields a lower bound on the PDE lifespan, because it keeps the solution bounded until the barrier's own blow-up time. A lower barrier yields an upper bound on the PDE lifespan. Reversing those conclusions is a common error. A numerical time integrator that overflows or stops at a large value does not establish the mathematical Tmax. One needs an analytic comparison or another validated criterion and must distinguish amplitude growth from a discretization instability.

For periodic boundary conditions, the average of the Laplacian is zero. If m(t) is the spatial mean, then m′ is the mean of uᵖ. Convexity gives m′≥mᵖ for a nonnegative solution. Positive initial mean therefore produces an ODE lower bound and forces finite-time amplitude blow-up no later than its associated ODE time. This conclusion uses a domain with no net diffusive loss. Homogeneous Neumann boundaries admit the same integration argument; absorbing Dirichlet boundaries introduce flux and can change the outcome.

Continuation is a logical alternative, not a claim that every local solution actually blows up. Zero data remain zero, and reactions with saturation can admit global bounded evolution. Even for a superlinear source, whole-space or absorbing-boundary questions can involve dimension, decay, and initial size in ways absent from the compact periodic argument. State the equation and boundary mechanism before drawing a global conclusion. A finite maximal time verified by an amplitude criterion is different from loss of a convenient explicit formula or failure of one approximation method.