Institutos Universitarios

Nonlinear PDEs and Applications

Brief description

Most of the relevant models in Applied Mathematics in connection with other Sciences and Engineering are formulated in terms of Nonlinear Partial Differential Equations (PDEs). The interdisciplinary treatment of many complex problems require to build different models which allow to have predictions previous or simultaneous to experimental studies. The spectrum of problems formulated in terms on nonlinear PDEs is very large and cover dynamical frameworks as well as their associated stationary regimes. Moreover, besides the direct approach involving the existence, uniqueness and regularity of solutions in suitable functional spaces well adapted to the model the mathematical treatment is completed with the consideration of inverse formulations and the control of the data in order to get desired properties to be satisfied by the solutions. In fact, the numerical approach also requires a suitable functional framework leading to convergent algorithms. Among the list of problems considered in this Program we mention the occurrence of strange terms in homogenization to critical scales, localizing formulations in linear and nonlinear Schrodinger equations, Reaction- Diffusion systems arising in Climatology and Chemical Engineering, Stochastic PDEs, Lithium batteries models, systems of equations in population studies, Image Processing and so on. In addition, the extension and generation of new mathematical techniques to deal with non-local formulations will be also carried out.




External Collaborators

  • Gregorio Díaz Díaz (Facultad de Matemáticas, UCM)



  • J. I. Díaz, J. Giacomoni. Monotone continuous dependence of solutions of singular quenching parabolic problems. Rendiconti del Circolo Matemático di Palermo Series 2, 2022, 72.
  • A. C. Casal, G. Díaz, J. I.Díaz, J. M. Vegas. Controlled boundary explosions: dynamics after blow-up for some semilinear problems with global controls. Discrete and Continuous Dynamical Systems Series S, 2023, 43(3&4).
  •  J. A. Carrillo, D. Gómez-Castro, Y. Yao, C. Zeng. Asymptotic Simplification of Aggregation-Diffusion Equations Towards the Heat kernel. Archive for Rational Mechanics and Analysis. 2023, 247(1).
  • Abatangelo, D. Gómez-Castro, J. L. Vázquez. Singular boundary behaviour and large solutions for fractional elliptic equations. Journal of the London Mathematical Society. 2023,
  • J. I. Díaz, A. Liñan. Descarga de gases en conductos largos: un problema de permanente actualidad. Boletín del IMI, Nº 63 (13 Oct 2022). Sección "1+400. Divulgación con 1 imagen y 400 palabras". 
  • J. I. Díaz, A. V. Podolskiy, T. A. Shaposhnikova. On the convergence of controls and cost functionals in some optimal control heterogeneous problems when the homogenization process gives rise to some strange terms. Journal of Mathematical Analysis and Applications. 2022, volumen 506, artículo 125559.

  • J. I. Díaz, A. V. Podolskiy, T. A. Shaposhnikova. On the homogenization of an optimal control problem in a domain perforated by holes of critical size and arbitrary shape. Doklady Mathematics, 2022, Vol.105, Nº 1, pp. 6-13. preprint
  • J. I. Díaz, J. Hernández, Bounded positive solutions for diffusive logistic equations with unbounded distributed limitations, Discrete & Continuous Dynamical Systems-S, 2022,
  • A. Carrillo, D. Gómez-Castro, J. Luis Vázquez, A fast regularisation of a Newtonian vortex equation, Annales de l'Institut Henri Poincaré C, 2022, vol. 39, nº3, pp.705-747
  • J. A. Carrillo, D. Gómez-Castro, J. L. Vázquez. Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility. Advances in Nonlinear Analysis. 2022, 11, 1, 937 – 967.
  • J. I. Díaz, A. V. Podolskiy, T. A. Shaposhnikova, Boundary control and homogenization: optimal climatization through smart double skin boundaries, Differential and Integral Equations, ISSN : 0893-4983, 2022, 35, 3 / 4, 191-210,
  • J. I. Díaz, A. V. Podolskiy, T. A. Shaposhnikova. On the homogenization of an optimal control problem in a domain perforated by holes of critical size and arbitrary shape. Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences). 2022, tomo 502, 11–18.
  • P. Bégout, J. I. Díaz. Finite time extinction for a class of damped Schrödinger equations with a singular saturated nonlinearity. Journal of Differential Equations. 2022, 308, 252-285.
  • J. I. Díaz, Formas geométricas naturales explicadas por la Física-Matemática: tréboles en aguas heladas, Boletín del IMI, Nº32 (13 de enero de 2022), Sección "1+400. Divulgación con 1 imagen y 400 palabras". link
  • G. Díaz, J. I. Díaz. Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing. Discrete and Continuous Dynamical Systems - Series S. 2021.
  • J. I. Díaz, J. Hernández. Multiple positive solutions for some local and non-local elliptic systems arising in desertification models. Rend. Mat. Appl. 2021, (7) 42 no. 3-4, 227–251. LinkPdf.
  • A. C. Casal, J. I. Díaz. Feedback Delay as a Control Tool: The Complex Ginzburg–Landau Equation with Local and Nonlocal Delayed Perturbations. In Recent Trends in Chaotic, Nonlinear and Complex Dynamics. Editorial: World Scientific (Series on Nonlinear Science Series B: Volume 19). 2021, 455 – 513. ISBN: 978-981-122-189-7 (hardcover), ISBN: 978-981-122-191-0 (eBook).
  • J. I. Díaz, D. Gómez-Castro, T. A. Shaposhnikova. Nonlinear Reaction-Diffusion Processes for Nanocomposites: Anomalous Improved Homogenization. De Gruyter Series in Nonlinear Analysis and Applications. 2021. Hardcover ISBN: 9783110647273, eBook ISBN: 9783110648997
  • N. A. Dao, J. I. DíazLogarithmically improved regularity criteria for the Navier-Stokes equations in homogeneous Besov spaces. Electronic Journal of Differential Equations. 2021, Article number 89. Link.
  • J. I. Díaz, F. Feo, M. R. Posteraro. Half-space Gaussian symmetrization: Applications to semilinear elliptic problems. Advances in Nonlinear Analysis. 2021, 10, 1, 1201-1221.
  • J. I. Díaz, D. Hilhorst, P. Kyriazopoulos. A parabolic system with strong absorption modeling dry-land vegetation. Electronic Journal of Differential Equations. 2021, 2021, Article number 8.
  • H. Chan, D. Gómez-Castro, J. L. Vázquez. Blow-up phenomena in nonlocal eigenvalue problems: When theories of L1 and L2 meet. Journal of Functional Analysis. 2021, 280, 7, 108845.
  • F. Brock, J. I. Díaz, A. Ferone, D. Gómez-Castro, A. Mercaldo. Steiner symmetrization for anisotropic quasilinear equations via partial discretization. Annales de l’Institut Henri Poincaré C, Analyse non linéaire. 2021, 38, 2, 347-368,
  • L. Brasco, D. Gómez-Castro, J. L. Vázquez. Characterisation of homogeneous fractional Sobolev spaces. Calculus of Variations and Partial Differential Equations. 2021, 60, 2, Article number 60.
  • N. A. Dao, J. I. Díaz, Q.-H. Nguyen. Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term. Advanced Nonlinear Studies. 2020, 20(2), 477-502.
  • V. Bobkov, P. Drábek, J. Hernández. Existence and multiplicity results for a class of semilinear elliptic equations. Nonlinear Analysis. 2020, 200, 112017.
  • J. I. Diaz, J. F. Padial, J. I. Tello, L. Tello. Complex Ginzburg-Landau equations with a delayed nonlocal perturbation. Electronic Journal of Differential Equations. 2020, 2020, No. 40, 1-18. ISSN: 1072-6691.
  • N. A. Dao, J. I. Díaz, Q.-H. Nguyen. Fractional Sobolev inequalities revisited: the maximal function approach. Rend. Lincei Mat. Appl. 2020, 31, 225–236.
  • N. A. Dao, J. I. Díaz. Energy and Large Time Estimates for Nonlinear Porous Medium Flow with Nonlocal Pressure in RN. Archive for Rational Mechanics and Analysis. 2020, 238 (1), 299-345.
  • F. del Teso, D. Gómez-Castro, J. L. Vázquez. Estimates on translations and Taylor expansions in fractional Sobolev spaces. Nonlinear Analysis. 2020, 200, 111995.
  • J. I. DíazD. Gómez-Castro, T. A. Shaposhnikova & M. N. Zubova. A Time-Dependent Strange Term Arising in Homogenization of an Elliptic Problem with Rapidly Alternating Neumann and Dynamic Boundary Conditions Specified at the Domain Boundary: The Critical Case. Doklady Mathematics. 2020, 101, 96–101.
  • P. Bégout, J. I. Díaz. Finite time extinction for the strongly damped nonlinear Schrödinger equation in bounded domains. Journal of Differential Equations. 2020, 268, 7, 4029-4058. 
  • J. I. DíazD. Gómez-Castro, A. V. Podolskiy, T. A. Shaposhnikova. Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes. Advances in Nonlinear Analysis. 2020, 9, 1, 193--227 doi:
  • J.I. DíazJ. Hernández, Y. Ilyasov. On the exact multiplicity of stable ground states of non-Lipschitz semi linear elliptic equations for some classes of star shaped sets. Advan. Nonlinear Anal. 2020, 9, 1046-1065. 



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  • 1 de diciembre de 2022. Tres miembros del IMI y tres miembros de su Comité Científico en los listados publicados por la Universidad de Stanford para dar a conocer los científicos con mayor número de citas a nivel mundial. Los listados incluyen a los 100.000 científicos con mayor c-score según métricas de SCOPUS (con y sin autocitas) o los que se encuentren entre el 2% de los más citados de su campo de investigación. El primer miembro del IMI en aparecer en los listados de citas a lo largo de una carrera, con datos de 1960 a 2021, es Juan Luis Vázquez Suárez, seguido de Julián López Gómez y Jesús Ildefonso Díaz Díaz. También aparecen en los mismos listados los miembros del Comité Científico Paul Rabinowitz, Simon Donaldson y Herbert Amann.

  • 29 de abril de 2022. David Gómez-Castro (miembro del Grupo MOMAT y del IMI) y Marc Jornet Sanz (Universitat de València) han sido galardonados con el Premio SeMA "Antonio Valle" al Joven Investigador 2022. Más información aquí. ¡Enhorabuena a los dos!