On nonnegative solutions of the parabolic differential inequality with $(p,q)$-Laplace on Riemannian manifolds

Abstract: In this paper, we establish Liouville-type theorems for parabolic differential inequalities with $(p,q)-$Laplacian operator on Riemannian manifolds. By a test function argument, we establish nonexistence results under suitable weighted volume growth assumptions involving potential. In particular, we can obtain nonexistence results for a wider class of parabolic inequalities.

• The paper introduces sharp Liouville-type nonexistence results for nonnegative weak solutions of parabolic differential inequalities.
• It employs innovative weighted volume growth conditions and refined test function methods to handle the (p,q)-Laplacian on noncompact Riemannian manifolds.
• The study extends classical Fujita thresholds to mixed nonlinear operators, highlighting implications for blow-up and global existence in nonlinear PDEs.

Liouville-Type Theorems for Parabolic Differential Inequalities with $(p,q)$-Laplacian on Riemannian Manifolds

Introduction

This paper addresses the classification of nonnegative solutions to a class of quasilinear parabolic inequalities on non-compact, complete Riemannian manifolds, involving the sum of $p$-Laplace and $q$-Laplace operators. The primary focus is on the inequality: $$\partial_t u - \Delta_p u - \Delta_q u \geq V(x,t) u^s$$ where $1 < q \leq p$, $s > \max\{1, p-1\}$, $V(x,t) > 0$ almost everywhere, and $(M,g)$ is a complete, non-compact Riemannian manifold. The work establishes sharp Liouville-type nonexistence results by introducing new weighted volume growth conditions, extending and generalizing a significant body of previous work for single Laplacian and $p$-Laplacian operators.

Context and Motivation

Liouville theorems are essential in the qualitative analysis of PDEs; they assert nonexistence of nontrivial nonnegative solutions under various conditions, constraining the global behavior of such PDEs. For parabolic PDEs, these theorems often hinge on the interplay between the growth of the underlying manifold, the exponent $s$, and the structure of the lower order (reaction) terms. While the classical Fujita phenomenon elucidates the transition between global existence and blow-up for reaction-diffusion equations in Euclidean space, analogous results on Riemannian manifolds, particularly for quasilinear and mixed operators (such as $p$0-Laplacian), require new analytical frameworks.

The $p$1-Laplacian arises naturally in models in nonlinear elasticity, variational problems with nonstandard growth, and mathematical physics. Its nonhomogeneous structure engenders analytical difficulties not present in the single $p$2-Laplacian case, necessitating refined potential-theoretic and test function techniques for nonexistence results.

Main Results

Statement of Nonexistence Theorems

The core technical achievements are two Liouville-type theorems (Theorems 1.5 and 1.6 in the paper) for nonnegative, weak solutions to the aforementioned inequality. The nonexistence is established under sharp weighted volume growth conditions on the potential $p$3 and the manifold geometry.

Hypotheses (HP1 and HP2)

Two principal volume growth assumptions, HP1 and HP2, are introduced. They involve precise upper bounds on weighted space-time integrals over cylindrical regions in $p$4, controlling the decay/growth of the potential relative to the intrinsic geometry. These are parametrized by exponents derived from scaling analysis of the equation, and are further refined to capture the dominant role of the lower-order $p$5-Laplacian in the critical case.

Strong claim: If either HP1 or HP2 is satisfied (with precisely quantified exponent and logarithmic bounds), then any nonnegative weak solution $p$6 with nonnegative initial datum in $p$7 must be identically zero a.e. on $p$8.

Canonical Examples and Thresholds

Explicit corollaries (Corollary 1.7) delineate the critical Fujita-type exponents for the prototypical case $p$9 and $q$0: for $q$1, all nonnegative weak solutions are trivial. This extends the classical Fujita threshold to anisotropic doubly nonlinear operators.

Furthermore, the paper provides extensions to nonconstant potentials of separated form $q$2, supplying sufficient integral conditions for nonexistence in terms of the $q$3-integrability of negative powers of $q$4 and $q$5 (Corollaries 1.8 and 1.9), with fully explicit algebraic and logarithmic scaling bounds.

Analytic Methodology

The proofs employ a sophisticated test function methodology paired with precise energy estimates, involving test functions of the form $q$6 for carefully tuned $q$7. The analysis leverages a delicate interplay between the nonlinear structure of the $q$8-Laplacian and the geometry-induced weighted volume conditions. The technical innovations include:

• Establishment of weighted integral inequalities for the solution and its gradients (key Lemmas 2.2, 2.3).
• A monotonicity argument and covering argument to reduce estimates over annuli in the space-time manifold to radial integral conditions.
• Singular auxiliary functions and optimal scaling choices, allowing exploitation of the lower-order $q$9-operator, which dominates in the critical scaling regime.

Such analysis generalizes previous comparison principles and test function strategies for the parabolic $$\partial_t u - \Delta_p u - \Delta_q u \geq V(x,t) u^s$$0-Laplacian to much broader operators, as seen in previous works [BBF, (Bhakta et al., 14 Oct 2025)], [MMP2], [VGM].

Comparison with Prior Work

The results subsume earlier theorems for the $$\partial_t u - \Delta_p u - \Delta_q u \geq V(x,t) u^s$$1-Laplacian and extend the techniques of Grigor'yan, Sun, and Verbitsky [AS, GSV], Mastrolia, Monticelli, and Punzo [MMP2], and others. Notably, the paper generalizes "critical" integral conditions from the elliptic setting to the parabolic $$\partial_t u - \Delta_p u - \Delta_q u \geq V(x,t) u^s$$2-Laplace scenario, treating both time and spatial variables on equal footing via weighted integral bounds.

The constructive counterexamples in the case that the weighted volume conditions fail, alluded to in the discussion, confirm sharpness of the nonlinear and geometric thresholds.

Implications and Future Directions

The explicit dependence of the results on the exponents $$\partial_t u - \Delta_p u - \Delta_q u \geq V(x,t) u^s$$3, and the manifold growth, underscores their optimality and applicability to a wide range of geometric settings. The theorems provide classification results that can be directly applied in the analysis of blow-up, global existence, and large-time asymptotics for nonlinear parabolic PDEs with mixed operators in variable media.

On the theoretical side, this framework invites further exploration of:

• Sharpness and attainability of the thresholds in more general geometric settings, especially non-doubling manifolds or with unbounded geometry.
• Extensions to systems or to equations with additional lower-order terms or nonlocal (fractional) analogs.
• Connections to stochastic processes, as the parabolic relations correspond to potential-theoretic properties of associated Markovian diffusions with non-homogeneous generators.
• Further generalization, as suggested by the final remark, to operators of the form $$\partial_t u - \Delta_p u - \Delta_q u \geq V(x,t) u^s$$4, encompassing models with non-standard growth.

The underlying techniques may also have implications for the regularity theory and quantitative estimates for gradient-dependent quasilinear equations on Riemannian or even sub-Riemannian manifolds.

Conclusion

This paper provides a comprehensive Liouville-type classification for nonnegative solutions to parabolic differential inequalities involving the $$\partial_t u - \Delta_p u - \Delta_q u \geq V(x,t) u^s$$5-Laplacian on general Riemannian manifolds. The results are predicated on sharp, weighted, geometry-dependent volume growth conditions of the underlying manifold and the potential. The methodology combines the test function method with refined integral criteria, leading to optimal nonexistence results that capture a wide array of geometric and analytic phenomena. The work constitutes a substantive advance in the qualitative analysis of nonlinear parabolic PDEs with non-homogeneous diffusion operators.