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Rigorous integration of semilinear parabolic PDEs: forward evolution bounds and projected boundary value problems

Jean-Philippe Lessard, Akitoshi Takayasu

arXiv:2610.29871v1math.NAcs.NAmath.DS

TL;DR

Rigorous validation of semilinear parabolic PDE trajectories must control error accumulation over long intervals and constrained endpoints. This paper develops improved evolution bounds for IVPs and projected BVPs, then applies them to three trajectories of the two-dimensional Ohta–Kawasaki equation. It proves global existence with eventual enclosures for two trajectories and validates a finite projected orbit segment, while stopping short of proving convergence to exact equilibria or a heteroclinic connection.

  • Problem

    Rigorous validation of semilinear parabolic PDE trajectories requires computable error bounds that control accumulation over long intervals, including when endpoints are constrained.

    Source: §1 · p. 1

  • Method

    The method improves semigroup-based evolution estimates for IVPs and projected BVPs, using forward variational integration and preserving mode information in time-step propagation.

    Source: §1 · pp. 1–2

  • Results

    For the two-dimensional Ohta–Kawasaki equation, two prescribed initial data yield global existence and eventual uniform enclosures near numerical stripe and spot approximations, and a third trajectory satisfies projected endpoint conditions.

    Source: §1 · p. 3, §7 · p. 32

  • Takeaways & Limitations

    The results provide quantitative information about validated trajectories, including global enclosures and a uniquely validated projected orbit segment.

    Source: §7 · pp. 32–33

    The results establish neither convergence to exact equilibria nor a heteroclinic connection; validated local invariant-manifold parameterizations are a proposed route toward the latter.

    Source: §7 · p. 33

Abstract

from arXiv · show

In this work, we present a rigorous integrator for initial and boundary value problems for semilinear parabolic PDEs. The method builds on earlier semigroup-based approaches while improving the bounds used in the fixed-point argument and extending the class of problems that can be treated. In particular, it controls the solution operator without requiring backward integration of the variational equation and allows projected boundary value problems to be studied through a simplified construction of an approximate inverse for the linearized operator. To illustrate the scope of the method, we present three applications to the two-dimensional Ohta--Kawasaki equation. For two initial value problems, we prove global existence and obtain explicit enclosures valid for all sufficiently large times around numerical approximations of stripe and spot equilibria. These results use a fixed-point formulation on an unbounded time interval, avoiding the construction of trapping regions. We also rigorously solve a projected boundary value problem with endpoint conditions chosen from linear approximations of the unstable and stable spaces of two saddle equilibria. This provides a step toward the rigorous study of heteroclinic saddle-to-saddle connections.

1 Introduction

The paper develops rigorous IVP and projected BVP methods for semilinear parabolic PDEs, addressing error accumulation over long intervals and constrained endpoints. Its Ohta–Kawasaki examples establish global existence with eventual enclosures for two trajectories and validate a finite projected orbit segment, without proving convergence to exact equilibria or a heteroclinic connection.

1 Introduction

  • Long-time validation must control accumulated errors, including for trajectories with constrained initial and terminal values.The applications address these two settings through unbounded-time fixed points and projected endpoint conditions.Source: §1 · pp. 1, 3
  • The method rigorously validates semilinear parabolic IVPs and projected BVPs by improving time-step error propagation, avoiding backward integration of the linearized equations, and retaining information lost by scalar bounds.The projected BVP inverse construction reduces to a finite matrix for endpoint parameters.Source: §1 · pp. 1–3
  • Finite Fourier modes are enclosed with interval arithmetic, while parabolic smoothing controls the infinite tail and permits derivatives in the nonlinear term to be treated directly.The modewise evolution bounds require only forward integration of the variational equation.Source: §1 · pp. 2–3
  • Multiplying finite propagator matrices before taking absolute values preserves cancellations between mode contributions that stepwise scalar bounds would lose.The interaction between finite modes and the infinite tail is bounded separately as a correction.Source: §1 · pp. 2–3
  • For global existence, the finite-time trajectory is joined to an unbounded-interval fixed point around an approximate equilibrium, with its residual included instead of a separate equilibrium proof or trapping region.Intermediate validated trajectories can provide an incoming enclosure small enough for the final contraction test.Source: §1 · p. 3
  • The projected BVP uses approximate unstable and stable endpoint spaces, an added scalar amplitude condition, and a finite-matrix inverse construction after eliminating the trajectory correction.The unstable endpoint dimension exceeds the target saddle’s numerical Morse index by one.Source: §1 · p. 3
  • Completing a connection to a saddle requires placing the terminal endpoint in its nonlinear stable manifold; validated local invariant-manifold parameterizations are identified as a further step.The paper’s projected endpoint conditions do not themselves establish a heteroclinic connection.Source: §1 · p. 3
  • Two Ohta–Kawasaki IVPs have proved global existence and eventual enclosures near stripe and spot approximations, while a third orbit satisfies projected endpoint conditions but proves no heteroclinic connection.The global results also do not establish convergence to exact equilibria.Source: §1 · pp. 3–4

2 Functional analytic setup

The setup represents periodic, coordinatewise-even PDE solutions by cosine Fourier coefficients and constructs a finite numerical approximation with an explicitly bounded tail. Weighted sequence and trajectory spaces support the infinite-dimensional formulation, while a fixed-point map based on the linearized evolution operator provides the validation framework.

2 Functional analytic setup

  • Under periodic boundary conditions and even symmetry in each coordinate, the PDE is represented by an infinite ODE system for cosine Fourier coefficients, with a finite-mode numerical approximation extended by zero and its nonlinear tail residual bounded.Finite truncations use Chebyshev time series, refined by Newton iteration.Source: §2 · pp. 5–6

2.1 Function spaces

  • The weighted ℓ1ν space makes Fourier series converge absolutely and uniformly, is a Banach algebra for convolution, and bounds spatial supremum error; the stronger trajectory norm takes each mode’s time supremum before summing.For derivative nonlinearities, a weaker sequence space tracks the frequency-weighted loss, which parabolic smoothing recovers after integration.Source: §2.1 · pp. 6–7

2.2 Infinite-dimensional ODEs on sequence space

  • The Fourier system separates the diagonal linear operator L, polynomial convolution nonlinearity N, and derivative multiplier Q, with a weaker sequence space accommodating the derivative loss in QN(u).Parabolic smoothing controls that loss in the mild formulation.Source: §2.2 · pp. 7–8

2.3 Fixed-point operator

  • Linearizing around the numerical trajectory yields an evolution-operator fixed-point map whose fixed points solve the IVP, and the vanishing derivative at the approximation supports contraction bounds.Long intervals are divided into time steps because direct bounds on the evolution operator can grow too large for the contraction test.Source: §2.3 · pp. 8–9

3 Non-autonomous semigroup approach for integrating PDEs

This section develops rigorous semigroup-based bounds for integrating PDE trajectories, including one-step solution enclosures and multi-step propagation. The estimates combine finite Fourier modes with the infinite tail and bound defect, derivative, and solution operators over time intervals.

3.1 A fixed-point formulation over several time steps

  • The recursive stepwise map produces a continuous solution from a fixed point, which is unique when the Radii Polynomial Theorem’s invariance and contraction conditions hold.Separate radii allow different error sizes on different steps.Source: §3.1 · pp. 10–11

3.2 Defect and derivative bounds

  • Initial-value propagation and integrated inhomogeneous contributions are bounded with separate operators, supplying the linear estimates needed for the defect and derivative bounds.The initial-value operator propagates data from a step endpoint; the inhomogeneous-term operators integrate a source on one step and propagate its contribution.Source: §3.2 · p. 12

3.2.1 The bound Ym

  • The defect bound accounts for both the differential residual within each step and endpoint mismatch with the preceding approximation or prescribed initial condition.Residual modes outside the numerical Fourier truncation must also be included.Source: §3.2.1 · pp. 12–13

3.2.2 The bound Zi

  • Derivative bounds estimate a local nonlinear change and propagate its effect to later observation steps, completing the contraction test once operator bounds are available.For the cubic nonlinearity N(z) = -z^3, the local bound is Zi(r) = 6∥¯ui∥Xir + 3r2.Source: §3.2.2 · p. 13

3.3 Linear propagation bounds

  • To prevent bounds from growing geometrically across time steps, the method preserves signed finite-dimensional propagator products before taking absolute values and treats finite-tail coupling as an additive correction.It bounds individual modes in a finite set and the infinite tail as a whole.Source: §3.3 · p. 14

3.3.1 One step

  • One-step estimates use forward fundamental solutions and explicit exponentials, avoiding backward integration; they combine finite-mode bounds with tail estimates that use semigroup smoothing.The finite-mode integral bound controls the inhomogeneous solution, while smoothing compensates for the nonlinear term’s loss of regularity in the tail.Source: §3.3.1 · pp. 15–16
  • The one-step construction gives a nonnegative inverse when the Schur complement is positive and uses it to bound the solution and its endpoint value.The stated bound applies when the relevant nonnegative condition and positive Schur-complement condition hold.Source: §3.3.1 · p. 16

3.3.2 Several steps

  • Signed finite-part products preserve Fourier-mode cancellations by postponing absolute values, while block-norm corrections handle finite-tail coupling; assembling products over M steps requires O(M^2) matrix multiplications.The one-step quantities are computed independently, and coupling corrections are assembled separately by recurrence.Source: §3.3.2 · pp. 17–18

3.3.3 The bounds W (Jm,τj) and V (Jm,τj)

  • The multi-step bounds propagate initial data and inhomogeneous terms across intervals while retaining per-Fourier-mode bounds for uncoupled products.Uncoupled products are multiplied by one-step matrices before block norms are taken; coupling corrections are bounded through block norms.Source: §3.3.3 · p. 18

4 Global existence via a fixed-point argument

The construction joins a validated finite-time trajectory to an unbounded-interval fixed point around an approximate stable equilibrium, adding validated steps when needed. Under the stated conditions, the resulting solution exists globally and has explicit enclosures on each segment and near the equilibrium.

4.1 The infinite step

  • The infinite-step fixed point is centered at an approximate stable equilibrium and includes its residual, so separate equilibrium validation and trapping-region construction are unnecessary.If the finite-time endpoint is too far from the equilibrium, added validated steps reduce the incoming bound until the infinite-step test holds.Source: §4 · p. 19, §4.1 · p. 21
  • For the infinite interval, linear bounds use stable finite modes and control the infinite tail; the fixed-point test is applied with the incoming endpoint held fixed.The approximate equilibrium need not be an exact equilibrium; its residual is included in the defect bound.Source: §4.1 · pp. 20–21

4.2 Validated intermediate steps

  • Intermediate intervals are validated sequentially from verified incoming endpoint enclosures, and their endpoint bounds are propagated until the infinite-step test is satisfied.The local test must hold uniformly over the incoming enclosure.Source: Figure 4 · p. 19, §4.2 · p. 22

4.3 Global enclosure

  • The validated finite-time trajectory, intermediate segments, and infinite-interval solution join to give a unique solution for all t ≥0 with explicit segmentwise and equilibrium-centered bounds.Uniqueness holds among trajectories satisfying the validation bounds.Source: §4.3 · p. 23

5 Boundary value problems projected onto linear spaces

The projected boundary value problem combines an initial-value trajectory map with unknown endpoint parameters constrained to finite-dimensional approximations of unstable and stable spaces. A reduced inverse and contraction bounds yield a rigorous existence result for a solution satisfying these projected conditions.

5.1 Endpoint conditions and the zero-finding problem

  • The endpoint formulation adds a scalar amplitude condition to balance the finite endpoint-matching equations against the finite endpoint parameters.The initial value is in an approximation of the trivial equilibrium’s unstable space; the terminal value is in an affine space centered at a numerical saddle with stable-space directions.Source: §5 · p. 23, §5.1 · pp. 24–25

5.2 Reduction of the linearized boundary value problem

  • The linearized BVP reduces to endpoint parameters, enabling an approximate inverse built from a finite Fourier block and a tail correction.The approximation omits the tail of the propagated initial directions, and the finite inverse residual is bounded using interval arithmetic.Source: §5.2 · pp. 25–26

5.3 The bound Y

  • The Y bound controls the Newton-like defect using the trajectory residual and a separate, sharper estimate of its terminal value.The inverse bound converts these residual estimates into componentwise bounds on the correction.Source: §5.3 · p. 27

5.4 The bound Z

  • The Z bound controls variation across the ball; its terminal-parameter contribution vanishes because the terminal parametrization is affine.The amplitude condition contributes terms depending on the initial-parameter radius.Source: §5.4 · p. 28

5.5 Existence theorem for the projected boundary value problem

  • Under the stated inverse and contraction bounds, the theorem proves a unique zero and a PDE solution satisfying the projected endpoint and amplitude conditions.The initial value lies in the prescribed unstable space, and the terminal value lies in the affine space centered at the numerical saddle approximation.Source: §5.5 · p. 28
  • For longer flight times, the implementation uses a multi-step trajectory map and product space while leaving the reduced inverse construction unchanged.Only the finite endpoint matrix needs numerical inversion; trajectory derivative bounds use the maximum of the stepwise row-sum bounds.Source: §5.5 · p. 29

6 Application to the 2D Ohta–Kawasaki equation

At ε = 0.39, the paper validates global stripe- and spot-pattern trajectories and a finite projected orbit between the origin and a saddle for the two-dimensional Ohta–Kawasaki equation. The global enclosures apply to two prescribed initial data; the projected orbit does not yet establish a heteroclinic connection.

6.1 Equilibria and the three trajectories

  • At ε = 0.39, the origin has four unstable directions, and the three trajectories describe stripe and spot approach and a projected orbit toward a saddle of Morse index three.The two IVP conclusions apply to the specified initial data, not every point near the origin.Source: §6 · p. 29, §6.1 · p. 30

6 Application to the 2D Ohta–Kawasaki equation

  • The two prescribed initial data yield global solutions with eventual uniform enclosures around approximate stripe and spot equilibria, while the third result is a finite projected orbit segment.The global enclosures use an unbounded-interval fixed-point argument; they do not by themselves prove convergence to exact equilibria.Source: §6.2 · p. 30, §6.4 · p. 32

6.2 First trajectory: global existence near the stripe pattern

  • The stripe trajectory is validated for all t ≥ 0, with the infinite-step test succeeding after the finite-time validation ending at t = 13.No intermediate steps were needed before the infinite step; the enclosure is around an approximate stripe equilibrium and includes its residual.Source: §6.2 · pp. 30–31, §6.1 · p. 30

6.3 Second trajectory: a projected boundary value problem

  • The projected BVP has a unique validated zero and certifies a finite orbit with endpoints in prescribed projected affine spaces, not on the nonlinear invariant manifolds.A heteroclinic connection requires additional endpoint conditions.Source: §6.3 · pp. 31–32

6.4 Third trajectory: global existence near the spot pattern

  • The spot trajectory is validated for all t ≥ 0, using an intermediate chain before the infinite step; its numerical path passes near the (0, 2) saddle.The enclosure does not prove asymptotic convergence to an exact equilibrium; that requires an additional decay argument.Source: §6.4 · p. 32

7 Conclusion

The paper develops evolution-operator estimates that retain Fourier-mode information and support finite-time validation, global enclosures, and projected boundary value problems. Applications to the two-dimensional Ohta–Kawasaki equation establish global existence for two initial data and validate a projected orbit segment, while leaving convergence to exact equilibria and heteroclinic connections open.

7 Conclusion

  • Forward variational integration, semigroup comparison, and parabolic smoothing yield modewise error estimates that preserve cancellations and support finite-time trajectories, global enclosures, and projected BVPs.For projected BVPs, constructing the linearized inverse reduces to a finite system for endpoint parameters.Source: §7 · p. 32
  • Two Ohta–Kawasaki initial-value trajectories have global existence proofs and explicit uniform enclosures around numerical stripe and spot approximations for every time beyond certified entry times.When a direct connection to the infinite-step test is unavailable, an intermediate chain links it to the finite-time enclosure.Source: §7 · p. 32
  • The projected Ohta–Kawasaki orbit segment satisfies endpoint conditions associated with the origin and a numerical saddle, with uniqueness in an explicit product ball for endpoint parameters and trajectory.It does not establish convergence to exact equilibria or a heteroclinic connection.Source: §7 · p. 33
  • Validated equilibrium and decay estimates could help establish convergence to exact equilibria, while validated local manifold parameterizations could provide a route to heteroclinic connections.Affine arithmetic is also proposed as a way to reduce wrapping and improve bounds over longer integration intervals.Source: §7 · p. 33

A Newton refinement of the approximate solution

The approximate orbit is refined on each time step by Newton iteration in Chebyshev–Fourier coefficients. Its finite-dimensional residual and initial-condition equation define the coefficients used in validation.

A Newton refinement of the approximate solution

  • The finite-dimensional map combines differential-equation residual coefficients with a bordering equation imposing the current step’s initial condition.The resulting IVP system has n|F_N| equations for n|F_N| Chebyshev–Fourier coefficients.Source: A Newton refinement of the approximate… · p. 36
  • The derivative of the finite-dimensional map is assembled from the Chebyshev differentiation matrix, the diagonal Fourier multiplier, and the bordered initial-condition row.The step begins from the prescribed initial condition or an approximation of the preceding step’s endpoint, with mismatch included in the defect bound.Source: A Newton refinement of the approximate… · p. 36
  • Newton iteration refines the truncated-system solution from a Chebyshev interpolation of the numerical orbit, producing the approximate orbit used in validation.Iterations continue until the finite-dimensional residual is sufficiently small; the resulting approximation is validated by the Radii Polynomial Theorem.Source: A Newton refinement of the approximate… · pp. 36–37

B Details of the finite-time validation

The appendix provides proofs and tail constants supporting the finite-time validation argument; the main argument uses their conclusions.

B Details of the finite-time validation

B.1 Proof of the Radii Polynomial Theorem

The proof establishes a unique fixed point by showing that the map preserves a product ball and contracts in a weighted norm. For the Ohta–Kawasaki operator, tail estimates use positivity and monotonicity to bound infinite-time and finite-interval contributions.

B.1 Proof of the Radii Polynomial Theorem

  • A complete product ball is mapped into itself and contracted in a weighted norm, so the Banach fixed-point theorem gives a unique trajectory satisfying the prescribed initial condition.The componentwise bounds control the distance from the approximate trajectory.Source: §B.1 · pp. 37–38

B.2 Tail constants for the Ohta–Kawasaki operator

  • For Ohta–Kawasaki tail modes, positivity and monotonicity of δ bound infinite-time constants by 1/δ(r_K) and r_K/δ(r_K).For finite intervals, finitely many modes are evaluated directly and the remaining modes are bounded using r_N′/δ(r_N′); a finite-mode maximum alone is insufficient.Source: §B.2 · p. 38

B.3 Proof of the multi-step bounds

The multi-step bounds combine one-step remainders with blockwise product estimates to bound the linear solution and forcing across multiple intervals. The appendix identifies these linear bounds as inputs to the global-existence construction.

B.3 Proof of the multi-step bounds

  • The multi-step estimate combines the bounded product of one-step propagators with a separately bounded remainder from variation of constants.The proof applies block norms to signed products before taking bounds, then uses subadditivity and submultiplicativity in a telescoping argument.Source: §B.3 · pp. 38–39
  • The resulting operator-norm bound controls each interval’s solution by its initial data and forcing contributions.The proof obtains the bound by applying induced matrix 1-norms to the finite-dimensional block estimates.Source: §B.3 · p. 39

C Details of the global existence construction

C.1 Autonomous linear bounds

The autonomous estimates bound the semigroup and its forcing integrals on an infinite time interval by separating finite modes from the tail and then accounting for their coupling. A mode-resolved residual estimate can sharpen the residual contribution used in later radii tests.

C.1 Autonomous linear bounds

  • The uncoupled exponential integrals require negative real parts for the approximate finite-block eigenvalues and μK < 0, plus validated diagonalization and coupling errors.Source: §C.1 · p. 39
  • Finite-block and tail estimates are coupled through a verified positive Schur complement to bound the full linearized problem.The finite block is treated separately from the diagonal tail, with off-diagonal interactions and tail self-interaction included in the coupling.Source: §C.1 · p. 41
  • The infinite-time bounds control the homogeneous semigroup and forcing integrals both with and without the derivative multiplier Q.The estimates are first obtained on finite intervals with constants independent of the interval length, then extended by taking T →∞.Source: §C.1 · pp. 40–41
  • A small truncation can leave slowly decaying modes in the tail and prevent verification of s∞ > 0, so the selected truncation must be checked.Increasing the truncation moves those modes into the finite block.Source: §C.1 · p. 42

C.2 Mode-resolved equilibrium residual

  • Modewise tail integrals can replace the scalar residual term with a smaller verified residual bound in the radii tests.For a finite autonomous interval, the modewise factor is (1 − e^μkha)/(−μk).Source: §C.2 · p. 42

C.3 Intermediate linear and endpoint bounds

The intermediate-step construction gives verified flow and endpoint enclosures using finite-mode and tail bounds. It propagates incoming endpoint errors and residuals without backward integration or inversion of a cumulative endpoint product.

C.3 Intermediate linear and endpoint bounds

  • Moving slowly decaying modes into the intermediate finite block can reduce tail integrals, but the selected block must satisfy linear, positivity, and nonlinear radii conditions.Source: §C.3 · p. 43
  • The construction verifies a finite flow without backward integration of the variational equation.A verified Chebyshev fundamental solution can be used instead; for a constant reference and equal step lengths, the quantities can be computed once and reused.Source: §C.3 · p. 43
  • The endpoint enclosure propagates the incoming finite-tail error budget, nominal-point gap, residual forcing, and nonlinear integral across successive steps.The tail component of the incoming error is retained rather than set to zero.Source: §C.3 · pp. 44–45
  • No inverse of a cumulative endpoint product is used in the initial-error propagation.The signed interval product is evaluated before taking its magnitude bound.Source: §C.3 · p. 44
  • The affine endpoint enclosure holds when the local conditions are uniform, the contraction bound is below one, and the associated Schur complement is positive.Source: §C.3 · p. 45

C.4 Matching thresholds and numerical predictors

The matching thresholds balance incoming residuals against the available nonlinear margin, while numerical predictors guide step choices and initial radii. Predictor tests do not replace the verified nonlinear and endpoint-enclosure conditions.

C.4 Matching thresholds and numerical predictors

  • Heuristic quantities guide intermediate-step counts, lengths, and initial radii, but accepted choices must pass the rigorous inequalities and include incoming endpoint perturbations.Source: §C.4 · pp. 46–47
  • The matching threshold expresses the balance between the incoming residual and the available nonlinear margin.A positive threshold requires the condition stated for the threshold in the subsection.Source: §C.4 · p. 46
  • For cubic nonlinearities, a quadratic approximation can provide an initial radius guess, but validation uses the full cubic inequality.The full cubic has no greater margin than the quadratic bound, and validation also requires Z∞(r∞) < 1.Source: §C.4 · p. 46
  • The constant-center linear residual recurrence does not enclose the nonlinear trajectory.Source: §C.4 · p. 46

C.5 A product-space application of the Radii Polynomial Theorem

This appendix gives an alternative to sequential validation by applying Theorem 3.2 to a combined map on an extended product space. Its sufficient contraction and invariant-ball conditions yield the same global enclosure under (91) and (92).

C.5 A product-space application of the Radii Polynomial Theorem

  • Unlike sequential validation, contraction of the entire product map must include all incoming endpoint derivatives and defect bounds at the approximation of that map.Those defect bounds must include the recursive nominal endpoints σ(a).Source: §C.5 · pp. 47–48
  • Endpoint evaluation has norm at most one, making the resulting endpoint bounds valid.Source: §C.5 · p. 47
  • Under (91) and (92), applying Theorem 3.2 to the finite product, including its infinite-interval component, gives the same global enclosure as Corollary 4.3.Source: §C.5 · p. 48
  • Positive contraction weights exist when every diagonal derivative bound is below one, but the invariant-ball inequalities must still be checked.The derivative matrix is lower triangular; choosing all weights equal to one recovers the sufficient row-sum test.Source: §C.5 · p. 48
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