Pub Date : 2026-04-15Epub Date: 2026-01-16DOI: 10.1016/j.jde.2026.114115
Hilário Alencar , G. Pacelli Bessa , Gregório Silva Neto
In this paper, we establish nonexistence results for complete translating solitons of the r-mean curvature flow under suitable growth conditions on the -mean curvature and on the norm of the second fundamental form. We first show that such solitons cannot be entirely contained in the complement of a right rotational cone whose axis of symmetry is aligned with the translation direction. We then relax the growth condition on the -mean curvature and prove that properly immersed translating solitons cannot be confined to certain half-spaces opposite to the translation direction. We conclude the paper by showing that complete, properly immersed translating solitons satisfying appropriate growth conditions on the -mean curvature cannot lie completely within the intersection of two transversal vertical half-spaces.
{"title":"Half-space theorems for translating solitons of the r-mean curvature flow","authors":"Hilário Alencar , G. Pacelli Bessa , Gregório Silva Neto","doi":"10.1016/j.jde.2026.114115","DOIUrl":"10.1016/j.jde.2026.114115","url":null,"abstract":"<div><div>In this paper, we establish nonexistence results for complete translating solitons of the <em>r</em>-mean curvature flow under suitable growth conditions on the <span><math><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-mean curvature and on the norm of the second fundamental form. We first show that such solitons cannot be entirely contained in the complement of a right rotational cone whose axis of symmetry is aligned with the translation direction. We then relax the growth condition on the <span><math><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-mean curvature and prove that properly immersed translating solitons cannot be confined to certain half-spaces opposite to the translation direction. We conclude the paper by showing that complete, properly immersed translating solitons satisfying appropriate growth conditions on the <span><math><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-mean curvature cannot lie completely within the intersection of two transversal vertical half-spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114115"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate dispersive and Strichartz estimates for the Schrödinger equation involving the fractional Laplacian in real hyperbolic spaces and their discrete analogues, homogeneous trees. Due to the Knapp phenomenon, the Strichartz estimates on Euclidean spaces for the fractional Laplacian exhibit some loss of derivatives. A similar phenomenon appears on real hyperbolic spaces. However, such a loss disappears on homogeneous trees, due to the triviality of the estimates for small times.
{"title":"The Schrödinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees","authors":"Jean-Philippe Anker , Guendalina Palmirotta , Yannick Sire","doi":"10.1016/j.jde.2025.114065","DOIUrl":"10.1016/j.jde.2025.114065","url":null,"abstract":"<div><div>We investigate dispersive and Strichartz estimates for the Schrödinger equation involving the fractional Laplacian in real hyperbolic spaces and their discrete analogues, homogeneous trees. Due to the Knapp phenomenon, the Strichartz estimates on Euclidean spaces for the fractional Laplacian exhibit some loss of derivatives. A similar phenomenon appears on real hyperbolic spaces. However, such a loss disappears on homogeneous trees, due to the triviality of the estimates for small times.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114065"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2026-01-16DOI: 10.1016/j.jde.2026.114103
Tomás Caraballo , Alexandre N. Carvalho , Arthur C. Cunha , Heraclio López-Lázaro
In this paper, we introduce the concept of uniformly differentiable evolution processes for dynamical systems on families of time-dependent phase spaces. This framework is motivated by two main aspects: it provides an appropriate framework for studying the dynamics of solutions to non-cylindrical PDE problems, and it naturally extends the theory of uniformly differentiable evolution processes on fixed phase spaces. We establish sufficient conditions on the differential of the evolution process, decomposed as the sum of a contraction and an operator with compactness properties, ensuring that the associated pullback attractors have finite fractal dimension. Our approach is inspired by the smoothing property, Mañé's method, and techniques for controlling backward bounded trajectories. As an application, we analyze non-cylindrical problems with different geometries, studying the dynamics of solutions for the one-dimensional semilinear heat equation and for the two-dimensional Navier-Stokes equations.
{"title":"Smoothing property assumptions for uniformly differential processes acting on time-dependent normed spaces","authors":"Tomás Caraballo , Alexandre N. Carvalho , Arthur C. Cunha , Heraclio López-Lázaro","doi":"10.1016/j.jde.2026.114103","DOIUrl":"10.1016/j.jde.2026.114103","url":null,"abstract":"<div><div>In this paper, we introduce the concept of uniformly differentiable evolution processes for dynamical systems on families of time-dependent phase spaces. This framework is motivated by two main aspects: it provides an appropriate framework for studying the dynamics of solutions to non-cylindrical PDE problems, and it naturally extends the theory of uniformly differentiable evolution processes on fixed phase spaces. We establish sufficient conditions on the differential of the evolution process, decomposed as the sum of a contraction and an operator with compactness properties, ensuring that the associated pullback attractors have finite fractal dimension. Our approach is inspired by the smoothing property, Mañé's method, and techniques for controlling backward bounded trajectories. As an application, we analyze non-cylindrical problems with different geometries, studying the dynamics of solutions for the one-dimensional semilinear heat equation and for the two-dimensional Navier-Stokes equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114103"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-30DOI: 10.1016/j.jde.2025.114073
Enrique Otárola, Daniel Quero, Matías Sasso
We consider a bilinear optimal control problem with pointwise tracking for a semilinear elliptic PDE in two and three dimensions. The control variable enters the PDE as a (reaction) coefficient and the cost functional contains point evaluations of the state variable. These point evaluations lead to an adjoint problem with a linear combination of Dirac measures as a forcing term. In Lipschitz domains, we derive the existence of optimal solutions and analyze first and necessary and sufficient second order optimality conditions. We also prove that every locally optimal control belongs to . Finally, assuming that the domain is a convex polygon, we prove that .
{"title":"A bilinear pointwise tracking optimal control problem for a semilinear elliptic PDE","authors":"Enrique Otárola, Daniel Quero, Matías Sasso","doi":"10.1016/j.jde.2025.114073","DOIUrl":"10.1016/j.jde.2025.114073","url":null,"abstract":"<div><div>We consider a bilinear optimal control problem with pointwise tracking for a semilinear elliptic PDE in two and three dimensions. The control variable enters the PDE as a (reaction) coefficient and the cost functional contains point evaluations of the state variable. These point evaluations lead to an adjoint problem with a linear combination of Dirac measures as a forcing term. In Lipschitz domains, we derive the existence of optimal solutions and analyze first and necessary and sufficient second order optimality conditions. We also prove that every locally optimal control <span><math><mover><mrow><mi>u</mi></mrow><mrow><mo>¯</mo></mrow></mover></math></span> belongs to <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>. Finally, assuming that the domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> is a convex polygon, we prove that <span><math><mover><mrow><mi>u</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114073"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-31DOI: 10.1016/j.jde.2025.114070
Ningning Zhu , Taishan Yi , Yuming Chen
Based on the structure of equations, we develop a cross iteration method to study the existence and propagation characteristics of solutions to a parabolic-elliptic Lotka-Volterra system. First, combining the cross iteration process with the method of upper and lower solutions and employing an approximation argument from above and below, we obtain the existence of classical solutions for the system. Then, by using the maximum principle and the iterative method of travelling wave maps, we establish asymptotic estimates for solutions to a time-dependent elliptic equation and a parabolic equation. Finally, with these estimates, we characterize the asymptotic propagation characteristics of nontrivial solutions to the parabolic-elliptic system. This explains the intrinsic mechanism of coexistence and extinction for competing species on limiting fast-slow time scales.
{"title":"Existence and propagation phenomena on solutions to a parabolic-elliptic Lotka-Volterra system","authors":"Ningning Zhu , Taishan Yi , Yuming Chen","doi":"10.1016/j.jde.2025.114070","DOIUrl":"10.1016/j.jde.2025.114070","url":null,"abstract":"<div><div>Based on the structure of equations, we develop a cross iteration method to study the existence and propagation characteristics of solutions to a parabolic-elliptic Lotka-Volterra system. First, combining the cross iteration process with the method of upper and lower solutions and employing an approximation argument from above and below, we obtain the existence of classical solutions for the system. Then, by using the maximum principle and the iterative method of travelling wave maps, we establish asymptotic estimates for solutions to a time-dependent elliptic equation and a parabolic equation. Finally, with these estimates, we characterize the asymptotic propagation characteristics of nontrivial solutions to the parabolic-elliptic system. This explains the intrinsic mechanism of coexistence and extinction for competing species on limiting fast-slow time scales.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114070"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2026-01-02DOI: 10.1016/j.jde.2025.114075
Umberto Guarnotta, Cristina Marcelli
Sufficient conditions for either existence or non-existence of traveling wave solutions for a general quasi-linear reaction-diffusion-convection equation, possibly highly degenerate or singular, with discontinuous coefficients are furnished. Under an additional hypothesis on the convection term, the set of admissible wave speeds is characterized in terms of the minimum wave speed, which is estimated through a double-sided bound.
{"title":"Traveling waves for highly degenerate and singular reaction-diffusion-advection equations with discontinuous coefficients","authors":"Umberto Guarnotta, Cristina Marcelli","doi":"10.1016/j.jde.2025.114075","DOIUrl":"10.1016/j.jde.2025.114075","url":null,"abstract":"<div><div>Sufficient conditions for either existence or non-existence of traveling wave solutions for a general quasi-linear reaction-diffusion-convection equation, possibly highly degenerate or singular, with discontinuous coefficients are furnished. Under an additional hypothesis on the convection term, the set of admissible wave speeds is characterized in terms of the minimum wave speed, which is estimated through a double-sided bound.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114075"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2026-01-05DOI: 10.1016/j.jde.2025.114079
Peter De Maesschalck , Renato Huzak , Otavio Henrique Perez
The main purpose of this paper is to study limit cycles of non-linear regularizations of planar piecewise smooth systems. We deal with a slow-fast Hopf point after non-linear regularization and blow-up. We give a simple criterion for the existence of limit cycles of canard type blue for a class of (non-linearly) regularized piecewise smooth systems, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of a piecewise linear center such that for any integer it has at least limit cycles, for a suitably chosen monotonic transition function . We prove a similar result for regularized codimension 1 invisible-invisible fold-fold singularities of type . Canard cycles of dodging layer are also considered, and we prove that there can be at most 2 limit cycles (born in a saddle-node bifurcation).
{"title":"Canard cycles of non-linearly regularized piecewise smooth vector fields","authors":"Peter De Maesschalck , Renato Huzak , Otavio Henrique Perez","doi":"10.1016/j.jde.2025.114079","DOIUrl":"10.1016/j.jde.2025.114079","url":null,"abstract":"<div><div>The main purpose of this paper is to study limit cycles of non-linear regularizations of planar piecewise smooth systems. We deal with a slow-fast Hopf point after non-linear regularization and blow-up. We give a simple criterion for the existence of limit cycles of canard type blue for a class of (non-linearly) regularized piecewise smooth systems, expressed in terms of zeros of the slow divergence integral. Using the criterion we can construct a quadratic regularization of a piecewise linear center such that for any integer <span><math><mi>k</mi><mo>></mo><mn>0</mn></math></span> it has at least <span><math><mi>k</mi><mo>+</mo><mn>1</mn></math></span> limit cycles, for a suitably chosen monotonic transition function <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span>. We prove a similar result for regularized codimension 1 invisible-invisible fold-fold singularities of type <span><math><mi>I</mi><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Canard cycles of dodging layer are also considered, and we prove that there can be at most 2 limit cycles (born in a saddle-node bifurcation).</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114079"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2026-01-16DOI: 10.1016/j.jde.2026.114111
Fucai Li , Houzhi Tang , Shuxing Zhang
The classical Fourier's law, which states that the heat flux is proportional to the temperature gradient, induces the paradox of infinite propagation speed for heat conduction. To accurately simulate the real physical process, the hyperbolic model of heat conduction named Cattaneo's law was proposed, which leads to the finite speed of heat propagation. A natural question is whether the large-time behavior of the heat flux for compressible flow would be different for these two laws. In this paper, we aim to address this question by studying the global well-posedness and the optimal time-decay rates of classical solutions to the compressible Navier-Stokes system with Cattaneo's law. By designing a new method, we obtain the optimal time-decay rates for the highest order derivatives of the heat flux, which cannot be derived for the system with Fourier's law by Matsumura and Nishida (1979) [25]. In this sense, our results first reveal the essential differences between the two laws.
{"title":"Global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction","authors":"Fucai Li , Houzhi Tang , Shuxing Zhang","doi":"10.1016/j.jde.2026.114111","DOIUrl":"10.1016/j.jde.2026.114111","url":null,"abstract":"<div><div>The classical Fourier's law, which states that the heat flux is proportional to the temperature gradient, induces the paradox of infinite propagation speed for heat conduction. To accurately simulate the real physical process, the hyperbolic model of heat conduction named Cattaneo's law was proposed, which leads to the finite speed of heat propagation. A natural question is whether the large-time behavior of the heat flux for compressible flow would be different for these two laws. In this paper, we aim to address this question by studying the global well-posedness and the optimal time-decay rates of classical solutions to the compressible Navier-Stokes system with Cattaneo's law. By designing a new method, we obtain the optimal time-decay rates for the highest order derivatives of the heat flux, which cannot be derived for the system with Fourier's law by Matsumura and Nishida (1979) <span><span>[25]</span></span>. In this sense, our results first reveal the essential differences between the two laws.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114111"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-30DOI: 10.1016/j.jde.2025.114072
Shangjiang Guo
In this paper, without establishing the Poincaré map, we employ Lyapunov-Schmidt procedure to investigate the one-codimensional bifurcations from the periodic orbits in delay differential equations, and obtain some important formulas giving the relevant coefficients for the determinations of bifurcation direction and stability of the bifurcating periodic solutions.
{"title":"Bifurcation from periodic solutions in delay differential equations","authors":"Shangjiang Guo","doi":"10.1016/j.jde.2025.114072","DOIUrl":"10.1016/j.jde.2025.114072","url":null,"abstract":"<div><div>In this paper, without establishing the Poincaré map, we employ Lyapunov-Schmidt procedure to investigate the one-codimensional bifurcations from the periodic orbits in delay differential equations, and obtain some important formulas giving the relevant coefficients for the determinations of bifurcation direction and stability of the bifurcating periodic solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114072"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145880556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-15Epub Date: 2025-12-30DOI: 10.1016/j.jde.2025.114055
A.F.M. ter Elst, M.F. Wong
Consider the elliptic operator on a bounded connected open set of class , where the are Hölder continuous of order κ and , subject to Robin boundary conditions , with is complex valued and . We show that the kernel of the semigroup generated by −A is differentiable in each variable and that the derivatives are Hölder continuous of order κ. Moreover, we prove Gaussian kernel bounds and Hölder Gaussian bounds for the derivatives of the kernel.
考虑有界连通开集Ω +κ类的Rd上的椭圆算子a =−∑k,l=1d∂kckl∂l−∑k=1d∂kbk+∑k=1dck∂k+c0，其中ckl，bk,ck∈Cκ（Ω，C）是κ阶连续的Hölder，且c0∈l∞（Ω，C），服从Robin边界条件∂νu+β tru =0，其中β∈Cκ（∂Ω，C）是复值，κ∈(0,1)。我们证明了−A生成的半群的核在每个变量上都是可微的，并且其导数是κ阶Hölder连续的。此外，我们还证明了高斯核界和Hölder核导数的高斯界。
{"title":"Differentiability and kernel estimates for Robin operators","authors":"A.F.M. ter Elst, M.F. Wong","doi":"10.1016/j.jde.2025.114055","DOIUrl":"10.1016/j.jde.2025.114055","url":null,"abstract":"<div><div>Consider the elliptic operator<span><span><span><math><mi>A</mi><mo>=</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>,</mo><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi></mrow></munderover><msub><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msub><mspace></mspace><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi><mi>l</mi></mrow></msub><mspace></mspace><msub><mrow><mo>∂</mo></mrow><mrow><mi>l</mi></mrow></msub><mo>−</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi></mrow></munderover><msub><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msub><mspace></mspace><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>d</mi></mrow></munderover><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mspace></mspace><msub><mrow><mo>∂</mo></mrow><mrow><mi>k</mi></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span></span></span> on a bounded connected open set <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>κ</mi></mrow></msup></math></span>, where the <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi><mi>l</mi></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>(</mo><mi>Ω</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> are Hölder continuous of order <em>κ</em> and <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub><mo>(</mo><mi>Ω</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span>, subject to Robin boundary conditions <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>ν</mi></mrow></msub><mi>u</mi><mo>+</mo><mi>β</mi><mspace></mspace><mrow><mrow><mi>Tr</mi></mrow><mspace></mspace></mrow><mspace></mspace><mi>u</mi><mo>=</mo><mn>0</mn></math></span>, with <span><math><mi>β</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>κ</mi></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>,</mo><mi>C</mi><mo>)</mo></math></span> is complex valued and <span><math><mi>κ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>. We show that the kernel of the semigroup generated by −<em>A</em> is differentiable in each variable and that the derivatives are Hölder continuous of order <em>κ</em>. Moreover, we prove Gaussian kernel bounds and Hölder Gaussian bounds for the derivatives of the kernel.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114055"},"PeriodicalIF":2.3,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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