Accurate estimations of Any Eigenpairs of N-th Order Linear Boundary Value Problems

[EN] This paper provides a method to bound and calculate any eigenvalues and eigenfunctions of n-th order boundary value problems with sign-regular kernels subject to two-point boundary conditions. The method is based on the selection of a particular type of cone for each eigenpair to be determined, the recursive application of the operator associated to the equivalent integral problem to functions belonging to such a cone, and the calculation of the Collatz-Wielandt numbers of the resulting functions.Almenar, P.; Jódar Sánchez, LA. (2021). Accurate estimations of Any Eigenpairs of N-th Order Linear Boundary Value Problems. Mathematics. 9(21):1-22. https://doi.org/10.3390/math921266312292

The sign of the Green function of an n-th order linear boundary value problem

[EN] This paper provides results on the sign of the Green function (and its partial derivatives) of ann-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The dependence of the absolute value of the Green function and some of its partial derivatives with respect to the extremes where the boundary conditions are set is also assessed.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Almenar, P.; Jódar Sánchez, LA. (2020). The sign of the Green function of an n-th order linear boundary value problem. Mathematics. 8(5):1-22. https://doi.org/10.3390/math8050673S12285Butler, G. ., & Erbe, L. . (1983). Integral comparison theorems and extremal points for linear differential equations. Journal of Differential Equations, 47(2), 214-226. doi:10.1016/0022-0396(83)90034-7Peterson, A. C. (1979). Green’s functions for focal type boundary value problems. Rocky Mountain Journal of Mathematics, 9(4). doi:10.1216/rmj-1979-9-4-721Peterson, A. C. (1980). Focal Green’s functions for fourth-order differential equations. Journal of Mathematical Analysis and Applications, 75(2), 602-610. doi:10.1016/0022-247x(80)90104-3Elias, U. (1980). Green’s functions for a non-disconjugate differential operator. Journal of Differential Equations, 37(3), 318-350. doi:10.1016/0022-0396(80)90103-5Nehari, Z. (1967). Disconjugate linear differential operators. Transactions of the American Mathematical Society, 129(3), 500-500. doi:10.1090/s0002-9947-1967-0219781-0Keener, M. S., & Travis, C. C. (1978). Positive Cones and Focal Points for a Class of nth Order Differential Equations. Transactions of the American Mathematical Society, 237, 331. doi:10.2307/1997625Schmitt, K., & Smith, H. L. (1978). Positive solutions and conjugate points for systems of differential equations. Nonlinear Analysis: Theory, Methods & Applications, 2(1), 93-105. doi:10.1016/0362-546x(78)90045-7Eloe, P. W., Hankerson, D., & Henderson, J. (1992). Positive solutions and conjugate points for multipoint boundary value problems. Journal of Differential Equations, 95(1), 20-32. doi:10.1016/0022-0396(92)90041-kEloe, P. W., & Henderson, J. (1994). Focal Point Characterizations and Comparisons for Right Focal Differential Operators. Journal of Mathematical Analysis and Applications, 181(1), 22-34. doi:10.1006/jmaa.1994.1003Almenar, P., & Jódar, L. (2015). Solvability ofNth Order Linear Boundary Value Problems. International Journal of Differential Equations, 2015, 1-19. doi:10.1155/2015/230405Almenar, P., & Jódar, L. (2016). Improving Results on Solvability of a Class ofnth-Order Linear Boundary Value Problems. International Journal of Differential Equations, 2016, 1-10. doi:10.1155/2016/3750530Almenar, P., & Jodar, L. (2017). SOLVABILITY OF A CLASS OF N -TH ORDER LINEAR FOCAL PROBLEMS. Mathematical Modelling and Analysis, 22(4), 528-547. doi:10.3846/13926292.2017.1329757Sun, Y., Sun, Q., & Zhang, X. (2014). Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem. Abstract and Applied Analysis, 2014, 1-7. doi:10.1155/2014/513051Hao, X., Liu, L., & Wu, Y. (2015). Iterative solution to singular nth-order nonlocal boundary value problems. Boundary Value Problems, 2015(1). doi:10.1186/s13661-015-0393-6Webb, J. R. L. (2017). New fixed point index results and nonlinear boundary value problems. Bulletin of the London Mathematical Society, 49(3), 534-547. doi:10.1112/blms.12055Jiang, D., & Yuan, C. (2010). The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Analysis: Theory, Methods & Applications, 72(2), 710-719. doi:10.1016/j.na.2009.07.012Wang, Y., & Liu, L. (2017). Positive properties of the Green function for two-term fractional differential equations and its application. The Journal of Nonlinear Sciences and Applications, 10(04), 2094-2102. doi:10.22436/jnsa.010.04.63Zhang, L., & Tian, H. (2017). Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations. Advances in Difference Equations, 2017(1). doi:10.1186/s13662-017-1157-7Wang, Y. (2020). The Green’s function of a class of two-term fractional differential equation boundary value problem and its applications. Advances in Difference Equations, 2020(1). doi:10.1186/s13662-020-02549-

The principal eigenvalue of some nth order linear boundary value problems

[EN] The purpose of this paper is to present a procedure for the estimation of the smallest eigenvalues and their associated eigenfunctions of nth order linear boundary value problems with homogeneous boundary conditions defined in terms of quasi-derivatives. The procedure is based on the iterative application of the equivalent integral operator to functions of a cone and the calculation of the Collatz-Wielandt numbers of such functions. Some results on the sign of the Green functions of the boundary value problems are also provided.Almenar, P.; Jódar Sánchez, LA. (2021). The principal eigenvalue of some nth order linear boundary value problems. Boundary Value Problems. 2021:1-16. https://doi.org/10.1186/s13661-021-01561-2S116202

New results on the sign of the Green function of a two-point n-th order linear boundary value problem

[EN] This paper provides conditions for determining the sign of all the partial derivatives of the Green functions of n-th order boundary value problems subject to a wide set of homogeneous two-point boundary conditions, removing restrictions of previous results about the distance between the two extremes that define the problem. To do so, it analyzes the sign of the derivatives of the solutions of related two-point n-th order boundary value problems subject to n ¿ 1 boundary conditions by introducing a new property denoted by `hyperdisfocality¿Almenar-Belenguer, P.; Jódar Sánchez, LA. (2022). New results on the sign of the Green function of a two-point n-th order linear boundary value problem. Boundary Value Problems. 1-22. https://doi.org/10.1186/s13661-022-01631-z12

Construcción de soluciones precisas de problemas mixtos para ecuaciones en derivadas parciales con coeficientes variables separables

En esta memoria se construyen soluciones analítico-numéricas con cotas de error a priori de problemas mixtos relacionados con ecuaciones en derivadas parciales con coeficientes separables y condiciones de contorno de tipo Dirichlet. Mediante estrategias de truncación de soluciones formales en serie, el métido de autofunciones y métodos multipaso, se construyen aproximaciones analítico-numéricas con un error prefijado en un dominio acotado. La ecuación de difusión, de ondas y del telégrafo con coeficientes variables son tratadas. La unicidad de cada uno de los problemas es analizada.Almenar Belenguer, P. (1999). Construcción de soluciones precisas de problemas mixtos para ecuaciones en derivadas parciales con coeficientes variables separables [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/6066Palanci

Solvability of Nth Order Linear Boundary Value Problems

Copyright © 2015 P. Almenar and L. Jódar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This paper presents a method that provides necessary and sufficient conditions for the existence of solutions of nth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2013-41765-P.Almenar, P.; Jódar Sánchez, LA. (2015). Solvability of Nth Order Linear Boundary Value Problems. International Journal of Differential Equations. 2015:1-19. https://doi.org/10.1155/2015/230405S1192015Almenar, P., & Jódar, L. (2014). The Distance between Points of a Solution of a Second Order Linear Differential Equation Satisfying General Boundary Conditions. Abstract and Applied Analysis, 2014, 1-17. doi:10.1155/2014/126713Greguš, M. (1987). Third Order Linear Differential Equations. doi:10.1007/978-94-009-3715-4Polya, G. (1922). On the Mean-Value Theorem Corresponding to a Given Linear Homogeneous Differential Equations. Transactions of the American Mathematical Society, 24(4), 312. doi:10.2307/1988819Sherman, T. (1965). Properties of solutions ofn-th order linear differential equations. Pacific Journal of Mathematics, 15(3), 1045-1060. doi:10.2140/pjm.1965.15.1045Muldowney, J. S. (1979). A Necessary and Sufficient Condition for Disfocality. Proceedings of the American Mathematical Society, 74(1), 49. doi:10.2307/2042104Nehari, Z. (1967). Disconjugate Linear Differential Operators. Transactions of the American Mathematical Society, 129(3), 500. doi:10.2307/1994604Ahmad, S., & Lazer, A. C. (1978). AnN-Dimensional Extension of the Sturm Separation and Comparison Theory to a Class of Nonselfadjoint Systems. SIAM Journal on Mathematical Analysis, 9(6), 1137-1150. doi:10.1137/0509092Ahmad, S., & Lazer, A. C. (1980). On nth-order Sturmian theory. Journal of Differential Equations, 35(1), 87-112. doi:10.1016/0022-0396(80)90051-0Elias, U. (1975). The extremal solutions of the equation Ly + p(x)y = 0. Journal of Mathematical Analysis and Applications, 50(3), 447-457. doi:10.1016/0022-247x(75)90001-3Elias, U. (1977). Nonoscillation and Eventual Disconjugacy. Proceedings of the American Mathematical Society, 66(2), 269. doi:10.2307/2040944Elias, U. (1978). Eigenvalue problems for the equation Ly + λp(x) y = 0. Journal of Differential Equations, 29(1), 28-57. doi:10.1016/0022-0396(78)90039-6Deimling, K. (1985). Nonlinear Functional Analysis. doi:10.1007/978-3-662-00547-7Gentry, R. D., & Travis, C. C. (1976). Comparison of Eigenvalues Associated With Linear Differential Equations of Arbitrary Order. Transactions of the American Mathematical Society, 223, 167. doi:10.2307/1997522Schmitt, K., & Smith, H. L. (1978). Positive solutions and conjugate points for systems of differential equations. Nonlinear Analysis: Theory, Methods & Applications, 2(1), 93-105. doi:10.1016/0362-546x(78)90045-7Keener, M. S., & Travis, C. C. (1978). Positive Cones and Focal Points for a Class of nth Order Differential Equations. Transactions of the American Mathematical Society, 237, 331. doi:10.2307/1997625Tomastik, E. C. (1983). Comparison Theorems for Second Order Nonselfadjoint Differential Systems. SIAM Journal on Mathematical Analysis, 14(1), 60-65. doi:10.1137/0514005Kreith, K. (1984). A class of hyperbolic focal point problems. Hiroshima Mathematical Journal, 14(1), 203-210. doi:10.32917/hmj/1206133155Hankerson, D., & Peterson, A. (1988). Comparison Theorems for Eigenvalue Problems for nth Order Differential Equations. Proceedings of the American Mathematical Society, 104(4), 1204. doi:10.2307/2047613Hankerson, D., & Henderson, J. (1990). Positive Solutions and Extremal Points for Differential Equations. Applicable Analysis, 39(2-3), 193-207. doi:10.1080/00036819008839980Eloe, P. W., Hankerson, D., & Henderson, J. (1992). Positive solutions and conjugate points for multipoint boundary value problems. Journal of Differential Equations, 95(1), 20-32. doi:10.1016/0022-0396(92)90041-kEloe, P. W., Hankerson, D., & Henderson, J. (1992). Positive Solutions and $J$-Focal Points for Two Point Boundary Value Problems. Rocky Mountain Journal of Mathematics, 22(4), 1283-1293. doi:10.1216/rmjm/1181072655Eloe, P. W., & Henderson, J. (1993). Focal Points and Comparison Theorems for a Class of Two Point Boundary Value Problems. Journal of Differential Equations, 103(2), 375-386. doi:10.1006/jdeq.1993.1055Eloe, P. W., & Henderson, J. (1994). Focal Point Characterizations and Comparisons for Right Focal Differential Operators. Journal of Mathematical Analysis and Applications, 181(1), 22-34. doi:10.1006/jmaa.1994.1003Eloe, P. ., Henderson, J., & Thompson, H. . (2000). Extremal points for impulsive Lidstone boundary value problems. Mathematical and Computer Modelling, 32(5-6), 687-698. doi:10.1016/s0895-7177(00)00165-5Eloe, P. W., & Ahmad, B. (2005). Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions. Applied Mathematics Letters, 18(5), 521-527. doi:10.1016/j.aml.2004.05.009Graef, J. R., & Yang, B. (2006). Positive solutions to a multi-point higher order boundary value problem. Journal of Mathematical Analysis and Applications, 316(2), 409-421. doi:10.1016/j.jmaa.2005.04.049Graef, J. R., Kong, L., & Wang, H. (2008). Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. Journal of Differential Equations, 245(5), 1185-1197. doi:10.1016/j.jde.2008.06.012Zhang, X., Feng, M., & Ge, W. (2009). Existence and nonexistence of positive solutions for a class of nth-order three-point boundary value problems in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 70(2), 584-597. doi:10.1016/j.na.2007.12.028Zhang, P. (2011). Iterative Solutions of Singular Boundary Value Problems of Third-Order Differential Equation. Boundary Value Problems, 2011, 1-10. doi:10.1155/2011/483057Sun, Y., Sun, Q., & Zhang, X. (2014). Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem. Abstract and Applied Analysis, 2014, 1-7. doi:10.1155/2014/513051Hao, X., Liu, L., & Wu, Y. (2015). Iterative solution to singular nth-order nonlocal boundary value problems. Boundary Value Problems, 2015(1). doi:10.1186/s13661-015-0393-6Lemmens, B., & Nussbaum, R. (2013). Continuity of the cone spectral radius. Proceedings of the American Mathematical Society, 141(8), 2741-2754. doi:10.1090/s0002-9939-2013-11520-0Eloe, P. W., & Ridenhour, J. (1994). Sign Properties of Green’s Functions for a Family of Two-Point Boundary Value Problems. Proceedings of the American Mathematical Society, 120(2), 443. doi:10.2307/215988

Convergent Disfocality and Nondisfocality Criteria for Second-Order Linear Differential Equations

Copyright © 2013 Pedro Almenar and Lucas Jódar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.This paper presents a method to determine whether the second-order linear differential equation y(n) + q(x)y = 0 is either disfocal or nondisfocal in a fixed interval. The method is based on the recursive application of a linear operator to certain functions and yields upper and lower bounds for the distances between a zero and its adjacent critical points, which will be shown to converge to the exact values of such distances as the recursivity index grows.This work has been supported by the Spanish Ministry of Science and Innovation Project DPI2010-C02-01.Almenar, P.; Jódar Sánchez, LA. (2013). Convergent Disfocality and Nondisfocality Criteria for Second-Order Linear Differential Equations. Abstract and Applied Analysis. 2013:1-11. doi:10.1155/2013/987976S1112013Kwong, M. K. (1981). On Lyapunov’s inequality for disfocality. Journal of Mathematical Analysis and Applications, 83(2), 486-494. doi:10.1016/0022-247x(81)90137-2Kwong, M. K. (1999). Integral Inequalities for Second-Order Linear Oscillation. Mathematical Inequalities & Applications, (1), 55-71. doi:10.7153/mia-02-06Harris, B. . (1990). On an inequality of Lyapunov for disfocality. Journal of Mathematical Analysis and Applications, 146(2), 495-500. doi:10.1016/0022-247x(90)90319-bBrown, R. C., & Hinton, D. B. (1997). Proceedings of the American Mathematical Society, 125(04), 1123-1130. doi:10.1090/s0002-9939-97-03907-5Tipler, F. J. (1978). General relativity and conjugate ordinary differential equations. Journal of Differential Equations, 30(2), 165-174. doi:10.1016/0022-0396(78)90012-8Došlý, O. (1993). Conjugacy Criteria for Second Order Differential Equations. Rocky Mountain Journal of Mathematics, 23(3), 849-861. doi:10.1216/rmjm/1181072527Moore, R. (1955). The behavior of solutions of a linear differential equation of second order. Pacific Journal of Mathematics, 5(1), 125-145. doi:10.2140/pjm.1955.5.125Almenar, P., & Jódar, L. (2012). An upper bound for the distance between a zero and a critical point of a solution of a second order linear differential equation. Computers & Mathematics with Applications, 63(1), 310-317. doi:10.1016/j.camwa.2011.11.023Almenar, P., & Jódar, L. (2013). The Distribution of Zeroes and Critical Points of Solutions of a Second Order Half-Linear Differential Equation. Abstract and Applied Analysis, 2013, 1-6. doi:10.1155/2013/147192Bellman, R. (1943). The stability of solutions of linear differential equations. Duke Mathematical Journal, 10(4), 643-647. doi:10.1215/s0012-7094-43-01059-

Improving Results on Solvability of a Class of nth-Order Linear Boundary Value Problems

Copyright © 2016 P. Almenar and L. Jodar. This is an open access article distributed under the Creative Commons Attribution ´ License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.[EN] This paper presents a modification of a recursive method described in a previous paper of the authors, which yields necessary and sufficient conditions for the existence of solutions of a class of �th-order linear boundary value problems, in the form of integral inequalities. Such a modification simplifies the assessment of the conditions on restricting the inequality to be verified to a single point instead of the full interval where the boundary value problem is defined. The paper also provides an error bound that needs to be considered in the integral inequalities of the previous paper when they are calculated numericallyThis work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2013-41765-P.Almenar, P.; Jódar Sánchez, LA. (2016). Improving Results on Solvability of a Class of nth-Order Linear Boundary Value Problems. International Journal of Differential Equations. https://doi.org/10.1155/2016/3750530S10Almenar, P., & Jódar, L. (2015). Solvability ofNth Order Linear Boundary Value Problems. International Journal of Differential Equations, 2015, 1-19. doi:10.1155/2015/230405Keener, M. S., & Travis, C. C. (1978). Positive Cones and Focal Points for a Class of nth Order Differential Equations. Transactions of the American Mathematical Society, 237, 331. doi:10.2307/1997625Gentry, R. D., & Travis, C. C. (1976). Comparison of eigenvalues associated with linear differential equations of arbitrary order. Transactions of the American Mathematical Society, 223, 167-167. doi:10.1090/s0002-9947-1976-0425241-xSchmitt, K., & Smith, H. L. (1978). Positive solutions and conjugate points for systems of differential equations. Nonlinear Analysis: Theory, Methods & Applications, 2(1), 93-105. doi:10.1016/0362-546x(78)90045-7Tomastik, E. C. (1983). Comparison Theorems for Second Order Nonselfadjoint Differential Systems. SIAM Journal on Mathematical Analysis, 14(1), 60-65. doi:10.1137/0514005Hankerson, D., & Henderson, J. (1990). Positive Solutions and Extremal Points for Differential Equations. Applicable Analysis, 39(2-3), 193-207. doi:10.1080/00036819008839980Eloe, P. W., Hankerson, D., & Henderson, J. (1992). Positive solutions and conjugate points for multipoint boundary value problems. Journal of Differential Equations, 95(1), 20-32. doi:10.1016/0022-0396(92)90041-kEloe, P. W., & Henderson, J. (1993). Focal Points and Comparison Theorems for a Class of Two Point Boundary Value Problems. Journal of Differential Equations, 103(2), 375-386. doi:10.1006/jdeq.1993.1055Eloe, P. W., & Henderson, J. (1994). Focal Point Characterizations and Comparisons for Right Focal Differential Operators. Journal of Mathematical Analysis and Applications, 181(1), 22-34. doi:10.1006/jmaa.1994.1003Eloe, P. ., Henderson, J., & Thompson, H. . (2000). Extremal points for impulsive Lidstone boundary value problems. Mathematical and Computer Modelling, 32(5-6), 687-698. doi:10.1016/s0895-7177(00)00165-5Eloe, P. W., & Ahmad, B. (2005). Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions. Applied Mathematics Letters, 18(5), 521-527. doi:10.1016/j.aml.2004.05.009Graef, J. R., & Yang, B. (2006). Positive solutions to a multi-point higher order boundary value problem. Journal of Mathematical Analysis and Applications, 316(2), 409-421. doi:10.1016/j.jmaa.2005.04.049Graef, J. R., Kong, L., & Wang, H. (2008). Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. Journal of Differential Equations, 245(5), 1185-1197. doi:10.1016/j.jde.2008.06.012Zhang, X., Feng, M., & Ge, W. (2009). Existence and nonexistence of positive solutions for a class of nth-order three-point boundary value problems in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications, 70(2), 584-597. doi:10.1016/j.na.2007.12.028Zhang, P. (2011). Iterative Solutions of Singular Boundary Value Problems of Third-Order Differential Equation. Boundary Value Problems, 2011, 1-10. doi:10.1155/2011/483057Sun, Y., Sun, Q., & Zhang, X. (2014). Existence and Nonexistence of Positive Solutions for a Higher-Order Three-Point Boundary Value Problem. Abstract and Applied Analysis, 2014, 1-7. doi:10.1155/2014/513051Hao, X., Liu, L., & Wu, Y. (2015). Iterative solution to singular nth-order nonlocal boundary value problems. Boundary Value Problems, 2015(1). doi:10.1186/s13661-015-0393-6Eloe, P. W., & Ridenhour, J. (1994). Sign Properties of Green’s Functions for a Family of Two-Point Boundary Value Problems. Proceedings of the American Mathematical Society, 120(2), 443. doi:10.2307/2159880Hämmerlin, G., & Hoffman, K.-H. (1991). Numerical Mathematics. Undergraduate Texts in Mathematics. doi:10.1007/978-1-4612-4442-

Shape Effect of Electrochemical Chloride Extraction in Structural Reinforced Concrete Elements Using a New Cement-Based Anodic System

This article shows the research carried out by the authors focused on how the shape of structural reinforced concrete elements treated with electrochemical chloride extraction can affect the efficiency of this process. Assuming the current use of different anode systems, the present study considers the comparison of results between conventional anodes based on Ti-RuO2 wire mesh and a cement-based anodic system such as a paste of graphite-cement. Reinforced concrete elements of a meter length were molded to serve as laboratory specimens, to closely represent authentic structural supports, with circular and rectangular sections. Results confirm almost equal performances for both types of anode systems when electrochemical chloride extraction is applied to isotropic structural elements. In the case of anisotropic ones, such as rectangular sections with no uniformly distributed rebar, differences in electrical flow density were detected during the treatment. Those differences were more extreme for Ti-RuO2 mesh anode system. This particular shape effect is evidenced by obtaining the efficiencies of electrochemical chloride extraction in different points of specimens.This research was funded by the Spanish Ministerio de Economía y Competitividad (and formerly by the Spanish Ministerio de Ciencia e Innovación) and ERDF (European Regional Development Fund) through projects BIA2010-20548 and MAT2009-10866, and also through the project PROMETEO/2013/035 of Generalitat Valenciana (Spain)

Early Detection of Corrosion-Induced Concrete Micro-cracking by Using Nonlinear Ultrasonic Techniques: Possible Influence of Mass Transport Processes

This work presents results allowing an unequivocal correlation of the observations of strong nonlinear elastic features of ultrasonic waves (values of the nonlinearity parameters exceeding the thresholds corresponding to undamaged states), with the critical events of the corrosion-induced surface cracking of reinforced cement-mortar specimens. These observations point to the possibility of the early detection of cracking using nonlinear ultrasonic (NLU) techniques. Experimental evidence is presented on the existence of active net mass transport processes, due to wick action, in the course of the corrosion tests, in the experimental conditions of this work. These phenomena might explain the observed abrupt shifting of the nonlinear parameter values (typically increasing and then decreasing post-peak, even reaching values typical of the undamaged state), and, partially, the high variability obtained for the values of the nonlinear parameters in damaged (cracked) states. Finally, some consequences are derived from the point of view of use of the NLU techniques in engineering practice, i.e., in surveys aimed at evaluating reinforced concrete structures affected by corrosion.This research was funded by the Spanish Agencia Estatal de Investigación (grant code BIA2016-80982-R) and by the European Regional Development Fund (grant code BIA2016-80982-R). M.M. acknowledges a pre-doctoral fellowship from the Spanish Ministerio de Educación, Cultura y Deporte (FPU16/04078)