Journal of Evolution Equations最新文献

{"title":"Three evolution problems modeling the interaction between acoustic waves and non-locally reacting surfaces","authors":"Enzo Vitillaro","doi":"10.1007/s00028-024-00974-7","DOIUrl":"https://doi.org/10.1007/s00028-024-00974-7","url":null,"abstract":"<p>The paper deals with three evolution problems arising in the physical modeling of small amplitude acoustic phenomena occurring in a fluid, bounded by a surface of extended reaction. The first one is the widely studied wave equation with acoustic boundary conditions, but its derivation from the physical model is mathematically not fully satisfactory. The other two models studied in the paper, in the Lagrangian and Eulerian settings, are physically transparent. In the paper the first model is derived from the other two in a rigorous way, also for solutions merely belonging to the natural energy spaces.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"15 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Second-order sufficient conditions in optimal control of evolution systems","authors":"H. Frankowska, E. M. Marchini, M. Mazzola","doi":"10.1007/s00028-024-00968-5","DOIUrl":"https://doi.org/10.1007/s00028-024-00968-5","url":null,"abstract":"<p>This paper concerns second-order sufficient conditions of optimality for a class of infinite dimensional optimal control problems under control constraints and end-point constraints. The distinctive feature of our formulation is the use of variational analysis techniques. Our approach is based on a suitable decomposition of controls, using second-order tangents, and it is developed in the case of functional constraints. An adaptation of the tools in the case of pointwise constraints on the controls is proposed too. We provide applications to concrete optimal control problems involving PDEs.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"175 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140808988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solvability of the Cauchy problem for fractional semilinear parabolic equations in critical and doubly critical cases","authors":"Yasuhito Miyamoto, Masamitsu Suzuki","doi":"10.1007/s00028-024-00967-6","DOIUrl":"https://doi.org/10.1007/s00028-024-00967-6","url":null,"abstract":"<p>Let <span>(0&lt;theta le 2)</span>, <span>(Nge 1)</span> and <span>(T&gt;0)</span>. We are concerned with the Cauchy problem for the fractional semilinear parabolic equation </p><span>$$begin{aligned} {left{ begin{array}{ll} partial _t u+(-Delta )^{theta /2}u=f(u) &amp;{} text {in} {{mathbb {R}}^N}times (0,T), u(x,0)=u_0 (x)ge 0 &amp;{} text {in} {{mathbb {R}}^N}. end{array}right. } end{aligned}$$</span><p>Here, <span>(fin C[0,infty ))</span> denotes a rather general growing nonlinearity and <span>(u_0)</span> may be unbounded. We study local in time solvability in the so-called critical and doubly critical cases. In particular, when <span>(f(u)=u^{1+{theta }/{N}}left[ log (u+e)right] ^{a})</span>, we obtain a sharp integrability condition on <span>(u_0)</span> which explicitly determines local in time existence/nonexistence of a nonnegative solution.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"69 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii type: the time-dependent case","authors":"Sebastian Grube","doi":"10.1007/s00028-024-00970-x","DOIUrl":"https://doi.org/10.1007/s00028-024-00970-x","url":null,"abstract":"<p>We consider a large class of nonlinear FPKEs with coefficients of Nemytskii type depending <i>explicitly</i> on time and space, for which it is known that there exists a sufficiently Sobolev-regular Schwartz-distributional solution <span>(uin L^1cap L^infty )</span>. We show that there exists a unique strong solution to the associated McKean–Vlasov SDE with time marginal law densities <i>u</i>. In particular, every weak solution of this equation with time marginal law densities <i>u</i> can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional produces a weak solution with time marginal law densities <i>u</i>.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"50 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Propagation of anisotropic Gabor singularities for Schrödinger type equations","authors":"","doi":"10.1007/s00028-024-00963-w","DOIUrl":"https://doi.org/10.1007/s00028-024-00963-w","url":null,"abstract":"<h3>Abstract</h3> <p>We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity <span> <span>(a(lambda x, lambda ^sigma xi ) = lambda ^{1+sigma } a(x,xi ))</span> </span> for <span> <span>(lambda &gt; 0)</span> </span> where <span> <span>(sigma &gt; 0)</span> </span> is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin–Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"298 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large deviation principle for multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motions","authors":"Guangjun Shen, Huan Zhou, Jiang-Lun Wu","doi":"10.1007/s00028-024-00960-z","DOIUrl":"https://doi.org/10.1007/s00028-024-00960-z","url":null,"abstract":"<p>In this paper, we are concerned with multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motion (with Hurst index <span>(H&gt;frac{1}{2})</span>) and standard Brownian motion, simultaneously. Our aim is to establish a large deviation principle for the multi-scale distribution-dependent stochastic differential equations. This is done via the weak convergence approach and our proof is based heavily on the fractional calculus.\u0000</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"16 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Perturbation of parabolic equations with time-dependent linear operators: convergence of linear processes and solutions","authors":"Maykel Belluzi","doi":"10.1007/s00028-024-00961-y","DOIUrl":"https://doi.org/10.1007/s00028-024-00961-y","url":null,"abstract":"<p>In this work, we consider parabolic equations of the form </p><span>$$begin{aligned} (u_{varepsilon })_t +A_{varepsilon }(t)u_{{varepsilon }} = F_{varepsilon } (t,u_{{varepsilon } }), end{aligned}$$</span><p>where <span>(varepsilon )</span> is a parameter in <span>([0,varepsilon _0))</span>, and <span>({A_{varepsilon }(t), tin {mathbb {R}}})</span> is a family of uniformly sectorial operators. As <span>(varepsilon rightarrow 0^{+})</span>, we assume that the equation converges to </p><span>$$begin{aligned} u_t +A_{0}(t)u_{} = F_{0} (t,u_{}). end{aligned}$$</span><p>The time-dependence found on the linear operators <span>(A_{varepsilon }(t))</span> implies that linear process is the central object to obtain solutions via variation of constants formula. Under suitable conditions on the family <span>(A_{varepsilon }(t))</span> and on its convergence to <span>(A_0(t))</span> when <span>(varepsilon rightarrow 0^{+})</span>, we obtain a Trotter-Kato type Approximation Theorem for the linear process <span>(U_{varepsilon }(t,tau ))</span> associated with <span>(A_{varepsilon }(t))</span>, estimating its convergence to the linear process <span>(U_0(t,tau ))</span> associated with <span>(A_0(t))</span>. Through the variation of constants formula and assuming that <span>(F_{varepsilon })</span> converges to <span>(F_0)</span>, we analyze how this linear process convergence is transferred to the solution of the semilinear equation. We illustrate the ideas in two examples. First a reaction-diffusion equation in a bounded smooth domain <span>(Omega subset {mathbb {R}}^{3})</span></p><span>$$begin{aligned}begin{aligned}&amp;(u_{varepsilon })_t - div (a_{varepsilon } (t,x) nabla u_{varepsilon }) +u_{varepsilon } = f_{varepsilon } (t,u_{varepsilon }), quad xin Omega , t&gt; tau , end{aligned} end{aligned}$$</span><p>where <span>(a_varepsilon )</span> converges to a function <span>(a_0)</span>, <span>(f_{varepsilon })</span> converges to <span>(f_0)</span>. We apply the abstract theory in this example, obtaining convergence of the linear process and solution. As a consequence, we also obtain upper-semicontinuity of the family of pullback attractors associated with each problem. The second example is a nonautonomous strongly damped wave equation </p><span>$$begin{aligned} u_{tt}+(-a(t) Delta _D) u + 2 (-a(t)Delta _D)^{frac{1}{2}} u_t = f(t,u), quad xin Omega , t&gt;tau ,end{aligned}$$</span><p>where <span>(Delta _D)</span> is the Laplacian operator with Dirichlet boundary conditions in a domain <span>(Omega )</span> and we analyze convergence of solution as we perturb the fractional powers of the associated linear operator.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"204 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sharp ill-posedness for the Hunter–Saxton equation on the line","authors":"","doi":"10.1007/s00028-024-00962-x","DOIUrl":"https://doi.org/10.1007/s00028-024-00962-x","url":null,"abstract":"<h3>Abstract</h3> <p>The purpose of this paper is to give an exact division for the well-posedness and ill-posedness (non-existence) of the Hunter–Saxton equation on the line. Since the force term is not bounded in the classical Besov spaces (even in the classical Sobolev spaces), a new mixed space <span> <span>(mathcal {B})</span> </span> is constructed to overcome this difficulty. More precisely, if the initial data <span> <span>(u_0in mathcal {B}cap dot{H}^{1}(mathbb {R}),)</span> </span> the local well-posedness of the Cauchy problem for the Hunter–Saxton equation is established in this space. Contrariwise, if <span> <span>(u_0in mathcal {B})</span> </span> but <span> <span>(u_0notin dot{H}^{1}(mathbb {R}),)</span> </span> the norm inflation and hence the ill-posedness is presented. It’s worth noting that this norm inflation occurs in the low frequency part, which exactly leads to a non-existence result. Moreover, the above result clarifies a corollary with physical significance such that all the smooth solutions in <span> <span>(L^{infty }(0,T;L^{infty }(mathbb {R})))</span> </span> must have the <span> <span>(dot{H}^1)</span> </span> norm.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"238 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571839","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces","authors":"Effie Papageorgiou","doi":"10.1007/s00028-024-00959-6","DOIUrl":"https://doi.org/10.1007/s00028-024-00959-6","url":null,"abstract":"<p>This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces <i>G</i>/<i>K</i> of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for <span>(L^1)</span> initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-<i>K</i>-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-<i>K</i>-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on <i>G</i>/<i>K</i>. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as <span>(L^1)</span> asymptotic convergence without the assumption of bi-<i>K</i>-invariance.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"7 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"KFP operators with coefficients measurable in time and Dini continuous in space","authors":"","doi":"10.1007/s00028-024-00964-9","DOIUrl":"https://doi.org/10.1007/s00028-024-00964-9","url":null,"abstract":"<h3>Abstract</h3> <p>We consider degenerate Kolmogorov–Fokker–Planck operators <span> <span>$$begin{aligned} mathcal {L}u&amp;=sum _{i,j=1}^{m_{0}}a_{ij}(x,t)partial _{x_{i}x_{j}} ^{2}u+sum _{k,j=1}^{N}b_{jk}x_{k}partial _{x_{j}}u-partial _{t}u&amp;equiv sum _{i,j=1}^{m_{0}}a_{ij}(x,t)partial _{x_{i}x_{j}}^{2}u+Yu end{aligned}$$</span> </span>(with <span> <span>((x,t)in mathbb {R}^{N+1})</span> </span> and <span> <span>(1le m_{0}le N)</span> </span>) such that the corresponding model operator having constant <span> <span>(a_{ij})</span> </span> is hypoelliptic, translation invariant w.r.t. a Lie group operation in <span> <span>(mathbb {R} ^{N+1})</span> </span> and 2-homogeneous w.r.t. a family of nonisotropic dilations. The matrix <span> <span>((a_{ij})_{i,j=1}^{m_{0}})</span> </span> is symmetric and uniformly positive on <span> <span>(mathbb {R}^{m_{0}})</span> </span>. The coefficients <span> <span>(a_{ij})</span> </span> are bounded and <em>Dini continuous in space</em>, and only bounded measurable in time. This means that, setting <span> <span>$$begin{aligned} mathrm {(i)}&amp;,,S_{T}=mathbb {R}^{N}times left( -infty ,Tright) , mathrm {(ii)}&amp;,,omega _{f,S_{T}}(r) = sup _{begin{array}{c} (x,t),(y,t)in S_{T} Vert x-yVert le r end{array}}vert f(x,t) -f(y,t)vert mathrm {(iii)}&amp;,,Vert fVert _{mathcal {D}( S_{T}) } =int _{0}^{1} frac{omega _{f,S_{T}}(r) }{r}dr+Vert fVert _{L^{infty }left( S_{T}right) } end{aligned}$$</span> </span>we require the finiteness of <span> <span>(Vert a_{ij}Vert _{mathcal {D}(S_{T})})</span> </span>. We bound <span> <span>(omega _{u_{x_{i}x_{j}},S_{T}})</span> </span>, <span> <span>(Vert u_{x_{i}x_{j}}Vert _{L^{infty }( S_{T}) })</span> </span> (<span> <span>(i,j=1,2,...,m_{0})</span> </span>), <span> <span>(omega _{Yu,S_{T}})</span> </span>, <span> <span>(Vert YuVert _{L^{infty }( S_{T}) })</span> </span> in terms of <span> <span>(omega _{mathcal {L}u,S_{T}})</span> </span>, <span> <span>(Vert mathcal {L}uVert _{L^{infty }( S_{T}) })</span> </span> and <span> <span>(Vert uVert _{L^{infty }left( S_{T}right) })</span> </span>, getting a control on the uniform continuity in space of <span> <span>(u_{x_{i}x_{j}},Yu)</span> </span> if <span> <span>(mathcal {L}u)</span> </span> is bounded and Dini-continuous in space. Under the additional assumption that both the coefficients <span> <span>(a_{ij})</span> </span> and <span> <span>(mathcal {L}u)</span> </span> are log-Dini continuous, meaning the finiteness of the quantity <span> <span>$$begin{aligned} int _{0}^{1}frac{omega _{f,S_{T}}left( rright) }{r}left| log rright| dr, end{aligned}$$</span> </span>we prove that <span> <span>(u_{x_{i}x_{j}})</span> </span> and <em>Yu</em> are Dini continuous; moreover, in this case, the derivatives <span> <span>(u_{x_{i}x_{j}})</span> </span> are locally uniformly continuous in space <em>and time</em>. </p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"4 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}