{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T18:08:41Z","timestamp":1788804521862,"version":"build-2803163510"},"reference-count":12,"publisher":"L and H Scientific Publishing, LLC","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["JVTSD"],"published-print":{"date-parts":[[2027,3,1]]},"abstract":"<jats:p>This paper investigates a class of malignant tumor spread models that balance the concentrations of invasive cells and the extracellular matrix. Using symmetry methods, invariant solutions are derived for the reaction-diffusion system governing tumor invasion. For a particular solution, numerical analysis reveals oscillatory spatial behavior governed by two parameters, with exponential growth in invasive cell concentration and linear growth in matrix degradation, which is characteristic of malignant tumors. New global existence results for classical solutions are established using fixed point theorems for the sum of two operators, proving the existence of one or more nonnegative solutions under suitable conditions. The approach combines the Kuratowski measure of noncompactness with expansive operator theory. A concrete mathematical example illustrates that the assumptions and conditions of the results can be satisfied.<\/jats:p>","DOI":"10.5890\/jvtsd.2027.03.007","type":"journal-article","created":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T17:41:02Z","timestamp":1788802862000},"page":"77-94","source":"Crossref","is-referenced-by-count":0,"title":["Malignant Tumours Spread Model Balancing Concentrations of the Invasive Cells and Extracellular Matrix"],"prefix":"10.5890","volume":"11","author":[{"given":"Ranis","family":"Ibragimov","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Svetlin G.","family":"Georgiev","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Georgette","family":"Owusu-Amankwah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hemanta","family":"Kalita","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Ntiamoah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"7015","published-online":{"date-parts":[[2026,9,7]]},"reference":[{"key":"ref1","doi-asserted-by":"crossref","unstructured":"[1] Perumpanani, A., Sherratt, J., Norbury, J., and Byrne, H. (1999), A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion, Physica D, 126, 145-159.","DOI":"10.1016\/S0167-2789(98)00272-3"},{"key":"ref2","doi-asserted-by":"crossref","unstructured":"[2] Aznavoorian, S., Stracke, M., Krutzsch, H., Schiffman, E., and Liotta, L. (1990), Signal transduction for chemotaxis and haptotaxis by matrix molecules in tumour cells, Journal of Cell Biology, 110, 1427-1438.","DOI":"10.1083\/jcb.110.4.1427"},{"key":"ref3","doi-asserted-by":"crossref","unstructured":"[3] Mignatti, P. and Rifkin, D. (1993), Biology and biochemistry of proteinases in tumor invasion, Physiological Reviews, 73, 161-195.","DOI":"10.1152\/physrev.1993.73.1.161"},{"key":"ref4","unstructured":"[4] Kolmogoroff, A., Petrovskii, I., and Piscounoff, N. (1937), Etude de l' equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique, Moscow Bulletin of Mathematics, 1, 1-25."},{"key":"ref5","doi-asserted-by":"crossref","unstructured":"[5] Sherratt, J. (1999), Traveling wave solutions of a mathematical model for tumor encapsulation, SIAM Journal on Applied Mathematics, 60(2), 392-407.","DOI":"10.1137\/S0036139998345355"},{"key":"ref6","doi-asserted-by":"crossref","unstructured":"[6] Deimling, K. (1985), Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg.","DOI":"10.1007\/978-3-662-00547-7"},{"key":"ref7","unstructured":"[7] Drabek, P. and Milota, J. (2007), Methods in Nonlinear Analysis, Applications to Differential Equations, Birkhauser."},{"key":"ref8","unstructured":"[8] Georgiev, S., Kheloufi, A., and Mebarki, K. (2024), Existence of classical solutions for a class of impulsive Hamilton-Jacobi equations, Palestine Journal of Mathematics, 13(4), 1084-1087."},{"key":"ref9","doi-asserted-by":"crossref","unstructured":"[9] Khemmar, K., Mebarki, K., and Georgiev, S. (2024), Existence of solutions for a class of boundary value problems for weighted p(t)-Laplacian impulsive systems, Filomat, 38, 7563-7577.","DOI":"10.2298\/FIL2421563K"},{"key":"ref10","unstructured":"[10] Djebali, S. and Mebarki, K. (2019), Fixed point index for expansive perturbation of k-set contraction mappings, Topological Methods in Nonlinear Analysis, 54(2), 613-640."},{"key":"ref11","doi-asserted-by":"crossref","unstructured":"[11] Mouhous, M., Georgiev, S., and Mebarki, K. (2022), Existence of solutions for a class of first order boundary value problems, Archivum Mathematicum, 58(3), 141-158.","DOI":"10.5817\/AM2022-3-141"},{"key":"ref12","doi-asserted-by":"crossref","unstructured":"[12] Polyanin, A. and Manzhirov, A. (1998), Handbook of Integral Equations, CRC Press.","DOI":"10.1201\/9781420050066"}],"container-title":["Journal of Vibration Testing and System Dynamics"],"original-title":[],"language":"en","deposited":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T17:44:47Z","timestamp":1788803087000},"score":1,"resource":{"primary":{"URL":"https:\/\/lhscientificpublishing.com\/index.php\/jvtsd\/article\/view\/2519"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,7]]},"references-count":12,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2027,3,1]]}},"URL":"https:\/\/doi.org\/10.5890\/jvtsd.2027.03.007","relation":{},"ISSN":["2475-4811","2475-482X"],"issn-type":[{"value":"2475-4811","type":"print"},{"value":"2475-482X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,7]]}}}