We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded L n ( p − 1 ) / 2 , ∞ ( R n ) -norm up to the blow-up time. As a consequence of this, we obtain optimal blow-up rates for certain radial solutions undergoing type II blow-up. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative ε -regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations.

Self-similar solutions of semilinear heat equations with positive speed

Non-positivity of the heat equation with non-local Robin boundary conditions

Fujita-type results for the semilinear heat equations driven by mixed local-nonlocal operators

Numerical study on characteristics of discrete motion and equivalent heat diffusion in granular shear flow

This paper investigates observer design and feedback stabilisation for a class of abstract boundary control systems. The proposed approach is based on two main steps. First, an auxiliary observer is constructed to estimate a lifted version of the system state, leading to a suitable observer-based feedback law for the closed-loop system. Second, a static reconstruction formula is introduced to recover the original state from the auxiliary estimate and the measured output. This yields an observer-based compensator ensuring the simultaneous exponential stabilisation of the plant and the observer. The main feature of the method is that the output injection acts in the internal observer dynamics while preserving the boundary control structure of the original system. Numerical simulations on both the one-dimensional and two-dimensional heat equations with Neumann boundary control are provided to demonstrate the effectiveness of the proposed framework. A numerical robustness assessment is further conducted to evaluate the closed-loop performance of the system in the presence of observer initialisation mismatch, exogenous disturbances, measurement noise, and moderate parametric uncertainty in the diffusion coefficient, providing encouraging numerical evidence of satisfactory closed-loop behaviour under these perturbation scenarios.

Abstract This work examines the free convection rotational flow of a second-grade (viscoelastic) fluid when a first-order chemical reaction and heat production are present. Such flows are important in marine and offshore engineering systems, where time-dependent boundary motion plays a significant role. To capture this behavior, the plate velocity is modeled using a generalized unsteady function f(t). The Laplace transform method is used to convert the governing equations for momentum, heat, and mass transport into dimensionless form and solve them analytically. The obtained results are also demonstrated graphically to see the effect of controlling parameters. The results indicate that viscoelastic effects enhance the near-wall velocity, whereas rotational effects suppress fluid motion. Stronger chemical reactions decrease the concentration field, whereas heat production broadens the temperature distribution and thickens the thermal boundary layer. It is also noticed that the heat transfer rate rises with heat generation, whereas the mass transfer rate decreases with increasing reaction rate. In addition, skin friction is found to increase with viscoelastic effects. In the limiting case( $$\alpha $$ -second grade parametr $$=0$$ ), the model reduces to the classical Newtonian fluid, confirming its validity. The present study provides useful insights into coupled heat and mass transfer in rotating non-Newtonian fluids, with potential applications in ocean engineering and marine energy systems.

In numerical methods, fine mesh sizes are often necessary to obtain highly accurate solutions of the differential equations and this increases the memory consumption and decreases the efficiency of the calculation. This paper presents a subdivision collocation algorithm of time-fractional advection diffusion equation that is a model that is used to characterize anomalous diffusion in scientific and engineering systems. The fractional time derivative is discretized in the Caputo sense to model memory effects, whereas spatial approximation is done using subdivision schemes. The approach converts the problem into an effective and steady system of algebraic equations by collocating at spatial nodes. The consistency and error analysis indicate that the methodology is reliable, and the numerical tests indicate that the suggested method is highly accurate and requires less computational tools than the current methods. Other than its numerical performance, the technique aids in real world simulation like the transportation of pollutants, heat conduction, and wave propagation in non-homogeneous media. The results highlight the subdivision collocation method as a promising tool for efficiently solving fractional partial differential equations while contributing to global sustainability challenges.

Within a unified framework, we establish the existence of general Poisson stable solutions for McKean–Vlasov stochastic differential equations. We show that if the coefficients are Poisson stable and satisfy appropriate conditions, then there exists a unique bounded solution that is globally asymptotically stable and preserves the same recurrence property as the coefficients. To illustrate our results, we provide an example of the stochastic heat equation.

Abstract We investigate density results for solutions of the non-local heat equation at a fixed time slice, considering two models: one with homogeneous Dirichlet boundary conditions and another with singular boundary data. In both cases, the non-local exponent satisfies $a \in (\tfrac{1}{2}, 1)$. We study both qualitative and quantitative approximation properties.
 
For the model with singular boundary data, we assume the potential is non-negative, sufficiently small, and exhibits mild growth. The smallness condition is explicit and depends only on the domain and the spatial dimension.
 
We also address Calderón-type inverse problems for these parabolic models, recovering the potential from solution data measured either on the boundary or at a fixed time slice. The Pohozaev identity is a key tool in establishing both the density results and the inverse problem analysis.
 
Finally, we apply the Pohozaev identity to an elliptic eigenvalue problem and show that the corresponding eigenfunctions, when divided by a suitable power of the distance to the boundary, cannot vanish on any non-empty open subset of the boundary. This result holds without restrictions on the non-local exponent.

Abstract With the advancement of tunnel and underground engineering, the geological environment has grown increasingly intricate, posing significant challenges to existing technologies. In particular, when tunnels traverse high geothermal areas, elevated temperatures within the tunnel not only compromise construction quality but also pose health risks to workers. Duct ventilation stands out as a widely employed mitigation measure. However, currently the temperature field variation induced by duct ventilation in construction tunnels can only be analysed through numerical simulations, requiring case-by-case modelling and computation, and the analytical assessment of the cooling efficacy of duct ventilation predominantly relies on steady-state analyses of operational tunnels, neglecting the consideration of the longitudinally differentiated distribution of heat transfer efficiency and the dynamic changes in the temperature field during tunnel construction. Motivated by these shortcomings, this study introduces a novel semi-analytical model for construction duct ventilation in high-thermal tunnels, utilizing the Green's function method (GFM) and Dirac's function. The model incorporates a trough equivalent ring heat source at the air-rock interface and a non-uniform convective heat transfer coefficient (CHTC) distributed along the longitudinal direction via a Gaussian function. This approach enables the determination of the unsteady temperature field within a dead-end tunnel, facilitating a comprehensive evaluation of the cooling effect. The findings reveal substantial variations in the temperature field of the surrounding rock along the longitudinal direction in a short timeframe, underscoring the significance of dynamic cooling assessments. The derived temperature distribution aligns well with numerical simulations and field-test results in high-geothermal tunnels. Further parametric analysis emphasizes the pronounced boundary effect of ventilation cooling, leading to suggested optimal ventilation rates tailored to different rock masses. This research provides valuable insights for optimizing tunnel construction in high-thermal environments.

Output regulation for a 1-D heat equation with infinite-Dimensional exosystem

Temperature variations may cause solid–liquid phase transition and damage evolution, which requires more refined modelling. In this study, a solid–liquid thermo-mechanical phase field model based on thermodynamically approach is established. A novel predictive equation of temperature-dependent critical strain energy density is firstly derived by combining force-heat equivalent principle and effective heat capacity method. The critical strain energy density of typical brittle materials with narrow phase transition interval (e.g., ice and Al 2 O 3 ) can be successfully captured by the proposed formulation. A simple but effective degradation function associated with phase transition variable is embedded in the phase field model to describe the mechanical degradation within phase transition and avoids the undesirable damage that occurs in liquid-state domain. The established multi-physical framework is implemented through finite element method. In numerical simulations, the phase transition part is verified through the two-phase Stefan’s melting issue preliminarily. Then, the thermo-mechanical module is studied through the shrinkage cracking of a 1D bar. The insensitivity of length scale and fracture toughness degradation under the assumption of small transition interval is proved. The proposed model is subsequently applied to thermal cracking in additive manufacturing and electrothermal de-icing, with its effectiveness and accuracy demonstrated by comparing with experimental results and empirical criterion

Thermal management of electronic chips using microencapsulated phase change material slurry in a taenidia-inspired spiral channel heat exchanger

Null controllability of the 1D heat equation with interior inverse square potential

Motivated by additive manufacturing (AM), we study surrogate modeling of transient heat conduction on complex geometries with a moving spot heat source. Existing surrogates are often limited to rectangular domains, require retraining for each new geometry, or rely on graph representations that add significant overhead and hinder scalability. To address these gaps, we introduce the Harmonic-Mapping Operator (HMO), a scalable, graph-free, and geometry-agnostic surrogate that learns thermo-physical responses without retraining for new part geometries. HMO maps arbitrary shapes onto a canonical square domain via harmonic maps, solves the transient heat equation with a single neural operator trained once, and projects results back through the inverse map. Geometry influences the model only through a metric tensor, enabling consistent generalization to unseen shapes. To scale to industrial components, a domain-decomposition inference (DDI) scheme applies the operator in parallel across subdomains while maintaining flux continuity. Compared with GeoFNO and MeshGraphNet baselines, HMO achieves lower errors, superior long-horizon stability, and faster inference, providing a reusable and real-time surrogate.

Repetitive learning output tracking control for a heat equation with multi-channel periodic disturbances

Quartic variation of the solution to the semilinear stochastic heat equation: Limit behavior and asymptotic independence with respect to the data

Theclassicalheatequationhasservedasacornerstoneofdiffusiontheoryandpartialdifferentialequationssince Fourier's seminal work in the early nineteenth century. Its elegant mathematical structure and predictive power in modeling thermal conduction and diffusion processes have made it indispensable across physics, engineering, and applied mathematics. However, experimental observations in complex media, including porous materials, biological tissues, and viscoelastic substances, reveal diffusion behaviors that deviate substantially from classical Fickian predictions. These anomalous transport phenomena exhibit memory effects, subdiffusion, and long-range spatial dependencies that cannot be adequately captured by integer-order differential operators. Fractional generalizations of the heat equation, incorporating fractional-order time derivatives, have emerged as natural extensions to model such nonlocal dynamics. This paper presents a comparative theoretical analysis of the classical heat equation and its fractional counterpart, examining their structural differences, solution properties, regularity characteristics, and physical interpretations. Weinvestigateexistenceanduniquenessresults, the nature ofregularity ineachframework, and the conceptual implications of introducing fractional derivatives into diffusion modeling. Rather than pursuing numerical experimentation, this study emphasizes analytical contrast and conceptual understanding, positioning the fractional heat equation not as a replacement but as a mathematically rigorous extension of classical theory with distinct applicability domains.

We consider the stochastic Yang-Mills heat equation on the two-dimensional torus. Using regularity structures, Chandra, Chevyrev, Hairer, and Shen previously proved both the local well-posedness and gauge-covariance of this model. In this article, we revisit their results using para-controlled calculus. One of the main ingredients is a new coordinate-invariant perspective on vector-valued stochastic objects.